History And Importance Of Algebra Mathematics Essay

In this project I will talk about starting of history of the algebra which is one of most important branches of arithmetic and Founder of the algebra and meaning of algebra and its benefit of our daily life, how we can learn and teach best way.
History of algebra
Algebra is an ancient and one of the most basic branches of mathematics. although inventor is Muhammad Musa Al-Khwarizmi, It was not developed or invented by a single person but it evolved over the centuries. The name algebra is itself of Arabic origin. It comes from the Arabic word ‘al-jebr’. The word was used in a book named ‘The Compendious Book on Calculation by Completion and Balancing’, written by the famous Persian mathematician Muhammad Musa al-Khwarizmi around 820 AD. Various derivations of the word “algebra,” which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title of a work by Mohammed Musa al-Khwarizmi , who flourished about the beginning of the 9th century. “The full title is ilm al-jebr wa’l-muqabala (algebra equations opposite) , means Science, which contains the ideas of restitution and comparison, or opposition and comparison or resolution and equation, jebr being derived from the verb jabara,  to reunite, and muqabala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a “bone-setter,” and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians.”1
Although the term “algebra” is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance.
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Algebra is one of the main areas of pure mathematics that uses mathematical statements such as term, equations, or expressions to relate relationships between objects that change over time. Many authors flourished algebra. by contributing specific field As well as
Cuthbert Tunstall
Cuthbert Tunstall (1474 -1559) was born in Hackforth, Yorkshire, England and died in Lambeth, London, England.
He was a significant royal advisor, diplomat, and administrator, and he gained two degrees with great proficiency in Greek, Latin, and mathematics.
In 1522, he wrote his first printed work that was devoted to mathematics, and this arithmetic book ‘De arte supputandi libri quattuor’ was based on Pacioli’s “Suma”.
Robert Recorde
1Robert Recorde (1510-1558) was born in Tenby, Wales and died in London, England.
He was a Welsh mathematician and physician and in 1557, he introduced the
equals sign (=).
In 1540, Recorde published the first English book of algebra ‘The Grounde of Artes’.
In 1557, he published another book ‘The Whetstone of Witte’ in which the equals sign was introduced.
John Widman
John Widman (1462-1498) was born in Eger, Bohemia, currently called Czech Republic and died in Leipzig, Germany.
He was a German mathematician who first introduced + and – signs in his arithmetic book ‘Behende und hupsche Rechnung auf Allen kauffmanschafft’.
How is Algebra used in daily life?
We use Algebra in finances, engineering, and many scientific fields. It is actually quite common for an average person to perform simple Algebra without realizing it. For example, if you go to the grocery store and have ten dollars to spend on two dollar candy bars. This gives us the equation 2x = 10 where x is the number of candy bars you can buy. Many people don’t realize that this sort of calculation is Algebra; they just do it.
1. http://wiki.answers.com/Q/Where_is_Algebra_used_in_daily_life#ixzz1KS594VsI
Basic laws of Algebra .
 There are five basic laws of algebra governing the operations of addition, subtraction, multiplication and division. And is expressed using the variables can be compensated for any number was. These laws are:
1 – substitution property of the collection. And write x + y = y + x. Means that the order is not important when collecting two issues as the result is the same. For example, 2 + 3 = 3 + 2 (-8) + (- 36) = (-36) + (-8).
2 – the property of the aggregate collection. And write C + (r + p) = (x + y) + p, which means that when you raise three issues or more, it can collect any form of first, and then complete the collection without affecting the final product, for example, 2 + (3 + 4) = (2 + 3) + 4 or 2 + 7 = 5 + 4.
3 – property substitution beaten. And write xy = y Q. Means that the order is not important when you hit the two issues as the result is the same. For example, (2) (3) = (3) (2) and (-8) (- 36) = (-36) (-8).
4 – aggregate property beaten. And write Q (r p) = (xy) p. Means that when you hit three or more numbers, it can hit any of them to form first, then complete the battery without affecting the final output. For example, 2 (3 Ã- 4) = (2 Ã- 3) 4 or 2 (12) = (6) 4.
5 – Distribution of property of multiplication over addition. And writes:
Q (r + p) = xy + x p.
Clarify this important property in algebra the following example:
3 (4 + 5) = (3 Ã- 4) + (3 Ã- 5). The multiplication of two numbers in the total number such as 3 (4 + 5) or 3 Ã- 9 equals the sum of multiplying the number one of the two numbers and multiplied by the number the second number. Note that:
3 (4 + 5) = 3 (9) = 27 as well.
(3 Ã- 4) + (3 Ã- 5) = 12 + 15 = 27.
Other definitions. It is important to know some other words used in algebra. Valmkdar o 2-2 XY + R contains three parts linked to the processes of addition or subtraction, called an end to every part of it. The amount of so-called compulsory component of the limit and only one Bouhid met, for example, 5 o r single limit, although it contains three elements (5, x, y) multiplied with each other and called each factor. And know how much that amount binomial component of their double-edged reference collection or ask, for example, both x + y, 3, a 2-4 with a double-edged. The polynomial is how much the component of the double-edged or more linked with each other or ask a reference collection, for example, Q – r + p polynomial. Note that the binomial is not only a special case of polynomial.

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That means the amounts set side by side in algebra they multiplied, Fidel expression on the 5 A product of a five-Issue 5 and is called a factor. Since that 5 times the symbol a in algebra is called a gradient of the number 5. As well as in the formula a (x + r) is a factor (x + y) and (x + y) is a factor. Since a = 1 Ã- a, we can always replace a formula 1a.
Combination. Similar to the process of bringing in algebra to a great extent than in the account. For example, the sum of A and A is 2a. We call a and 2 a similar double-edged because they contain the same variable. And to collect two quantities Ghebretin or more similar use property of the distribution of multiplication over addition, for example.
2x + 3 x + 4 h is (2 + 3 + 4) Q 9 or Q, but we can not express the sum of two quantities is similar with a single. For example, the sum of A and B written A + B. And to collect 3a, 4 b 0.6 a and b use his replacement and assembly of the collection process. It is clear that these special Tsaaadanna to collect any series of the border, written in any order. And the compilation of similar border, we find that:
3a +6 a = 9 a and 4 b + b b = 5.
So 3a +4 b + 6 a + b = 9 a + 5 b.
The solution could be organized as follows:
And to collect similar amounts of non-negative or positive, we were using a private distribution of multiplication over addition. To make it clear that use the collection:
(2a – b ² c + d 6 b ² + 2 d ) and
4 (a + 3 b ² c – 4 d b ² – 3 d ) and
3 (a + 2 b ² c + d 2 b ² – 4 d ) and
(-2 A – 8 b ² c + d 6 b ² + 6 d ).
And the number 3, which appears in the border such as 2 a means that a variable multiplied by itself three times. See: the cube. Before the process of collecting such amounts arrange the border in the columns.
Algebra equations
Algebra equation include letters represent unknown numbers.
It is one of the main branches of algebra in mathematics, where the mastery of mathematics depends on a proper understanding of algebra. And uses the engineers and scientists algebra every day, and counts commercial and industrial projects on the algebra to solve many of the dilemmas faced by them. Given the importance of algebra in modern life, it is taught in schools and universities all over the world.
Symbolizes the number of anonymous letters in algebra, such as X or Y. In some of the issues can be replaced only one number is indicated. As an example note that even a simple sentence becomes + 3 = 8 should be correct to compensate for x number 5 because 5 + 3 = 8.
In some other issues, it can compensate for the code number or more. For example, in order to achieve the health of sentence constraint x + y = 12 may put Q equals 6 and Y equals 6, or Q equal to 4, and Y equal to 8. In such sentences arrest, you can get several values ​​for x makes true if the sentences given for r different values.
And admire many of the students of his ability and usefulness of algebra big, as using algebra, one can solve many of the issues that can not be resolved by using the only account. For example, say the plane cut a distance of 1710 km in four hours if the flight in the direction of the wind blowing, but cut 1370 km in five hours if the flight was blowing the opposite direction of the wind. Using algebra, we can find the speed of the plane and wind speed.
Terminology used in algebra
Exponent of the number placed on the number or variable from the left to indicate the number of times where it is used as a factor.
Signals the assembly , brackets []. And are used in algebra formulas to account for arrest.
Square or second-degree variable multiplied by the same user as any ¸ twice •.
Binomial term in algebra consists of two double-edged symbol + or the symbol -.
The number of fixed or variable scope set of one item.
Roots of the equation numbers that make the equation correct a report when you replace the variables in the equation.
Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors. Moving from Arithmetic to Algebra will look like this: Arithmetic: 3 + 4 = 3 + 4 in Algebra it would look like: x + y = y + x
The name ‘algebra’ is derived from the treatise written by the Persian mathematician 
Muhammad bin MÅ«sā al-KhwārizmÄ« titled (in Arabic “Al-Kitab al-Jabr wa-l-Muqabala” 
The development of algebra is outlined in these notes under the following headings: Egyptian algebra, Babylonian algebra, Greek geometric algebra, Diophantine algebra, Hindu algebra, Arabic algebra, European algebra since 1500, and modern algebra.
Since algebra grows out of arithmetic, recognition of new numbers – irrationals, zero, negative numbers, and complex numbers – is an important part of its history.
And later became known to science in general mathematical equations
Best way to learn and teach algebra
As you already know, algebra is an essential subject. It’s the gateway to mathematics. It’s used extensively in the sciences. And it’s an important skill in many careers.
Yet for many people Algebra is a nightmare. It causes more stress, homework tears and plain confusion than any other subject on the curriculum.
Well the good news is you don’t have to struggle with Algebra for a minute longer. Because now there’s a solution that explains Algebra in a way that anyone can quickly understand.
Algebra is an Arabic word and a branch of mathematics and its name came from the book world of mathematics, astronomy and traveller Muhammad ibn Musa Khurazmi (short book, in the calculation of algebra and interview) which was submitted by the governing algebraic operations to find solutions to linear and quadratic equations.
The algebra is three branches of basic math in addition to geometry and mathematical analysis and the theory of numbers and permutations and combinations. And takes care of this science to study algebraic structures and symmetries, including, relations and quantities.
And algebra is the concept of a broader and more comprehensive account of the primary or reparation. It does not deal with numbers, but also formulate dealings with symbols, variables and categories as well. And formulate Alibdehyat algebra and relations by which can represent any phenomenon in the universe. So is one of the fundamentals governing the methods of proof
The Start of Algebra
Algebra is an ancient and one of the most basic branches of mathematics. It was not developed or invented by a single person but it evolved over the centuries. The name algebra is itself of Arabic origin. It comes from the Arabic word ‘al-jebr’. The word was used in a book named ‘The Compendious Book on Calculation by Completion and Balancing’, written by the famous Persian mathematician Muhammad ibn Musa ibn al-Khwarizmi around 820 AD. Various derivations of the word “algebra,” which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi), who flourished about the beginning of the 9th century. The full title is ilm al-jebr wa’l-muqabala, means Science, which contains the ideas of restitution and comparison, or opposition and comparison or resolution and equation, jebr being derived from the verb jabara, to reunite, and muqabala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a “bone-setter,” and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians.
Although the term “algebra” is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance.
Algebra is one of the main areas of pure mathematics that uses mathematical statements such as term, equations, or expressions to relate relationships between objects that change over time. Many authors flourished algebra. by contributing specific field As well as
Cuthbert Tunstall
Cuthbert Tunstall (1474 -1559) was born in Hackforth, Yorkshire, England and died in Lambeth, London, England.
He was a significant royal advisor, diplomat, and administrator, and he gained two degrees with great proficiency in Greek, Latin, and mathematics.
In 1522, he wrote his first printed work that was devoted to mathematics, and this arithmetic book ‘De arte supputandi libri quattuor’ was based on Pacioli’s “Suma”.
Robert Recorde
1Robert Recorde (1510-1558) was born in Tenby, Wales and died in London, England.
He was a Welsh mathematician and physician and in 1557, he introduced the equals sign (=).
In 1540, Recorde published the first English book of algebra ‘The Grounde of Artes’.
In 1557, he published another book ‘The Whetstone of Witte’ in which the equals sign was introduced.
John Widman
John Widman (1462-1498) was born in Eger, Bohemia, currently called Czech Republic and died in Leipzig, Germany.
He was a German mathematician who first introduced + and – signs in his arithmetic book ‘Behende und hupsche Rechnung auf Allen kauffmanschafft’.
How is Algebra used in daily life?
Mathematics is one of the first things you learn in life. Even as a baby you learn to count. Starting from that tiny age you will start to learn how to use building blocks how to count and then move on to drawing objects and figures. All of these things are important preparation to doing algebra..
We use Algebra in finances, engineering, and many scientific fields. It is actually quite common for an average person to perform simple Algebra without realizing it. For example, if you go to the grocery store and have ten dollars to spend on two dollar candy bars. This gives us the equation 2x = 10 where x is the number of candy bars you can buy. Many people don’t realize that this sort of calculation is Algebra; they just do it.2
———————-
2. http://wiki.answers.com/Q/Where_is_Algebra_used_in_daily_life#ixzz1KS594VsI
Basic laws of Algebra .
 There are five basic laws of algebra governing the operations of addition, subtraction, multiplication and division. And is expressed using the variables can be compensated for any number was. These laws are:
1 – substitution property of the collection. And write x + y = y + x. Means that the order is not important when collecting two issues as the result is the same. For example, 2 + 3 = 3 + 2 (-8) + (- 36) = (-36) + (-8).
2 – the property of the aggregate collection. And write C + (r + p) = (x + y) + p, which means that when you raise three issues or more, it can collect any form of first, and then complete the collection without affecting the final product, for example, 2 + (3 + 4) = (2 + 3) + 4 or 2 + 7 = 5 + 4.
3 – property substitution beaten. And write xy = y Q. Means that the order is not important when you hit the two issues as the result is the same. For example, (2) (3) = (3) (2) and (-8) (- 36) = (-36) (-8).
4 – aggregate property beaten. And write Q (r p) = (xy) p. Means that when you hit three or more numbers, it can hit any of them to form first, then complete the battery without affecting the final output. For example, 2 (3 Ã- 4) = (2 Ã- 3) 4 or 2 (12) = (6) 4.
5 – Distribution of property of multiplication over addition. And writes:
Q (r + p) = xy + x p.
Clarify this important property in algebra the following example:
3 (4 + 5) = (3 Ã- 4) + (3 Ã- 5). The multiplication of two numbers in the total number such as 3 (4 + 5) or 3 Ã- 9 equals the sum of multiplying the number one of the two numbers and multiplied by the number the second number. Note that:
3 (4 + 5) = 3 (9) = 27 as well.
(3 Ã- 4) + (3 Ã- 5) = 12 + 15 = 27.
Other definitions. It is important to know some other words used in algebra. Valmkdar o 2-2 XY + R contains three parts linked to the processes of addition or subtraction, called an end to every part of it. The amount of so-called compulsory component of the limit and only one Bouhid met, for example, 5 o r single limit, although it contains three elements (5, x, y) multiplied with each other and called each factor. And know how much that amount binomial component of their double-edged reference collection or ask, for example, both x + y, 3, a 2-4 with a double-edged. The polynomial is how much the component of the double-edged or more linked with each other or ask a reference collection, for example, Q – r + p polynomial. Note that the binomial is not only a special case of polynomial.
That means the amounts set side by side in algebra they multiplied, Fidel expression on the 5 A product of a five-Issue 5 and is called a factor. Since that 5 times the symbol a in algebra is called a gradient of the number 5. As well as in the formula a (x + r) is a factor (x + y) and (x + y) is a factor. Since a = 1 Ã- a, we can always replace a formula 1a.
Combination. Similar to the process of bringing in algebra to a great extent than in the account. For example, the sum of A and A is 2a. We call a and 2 a similar double-edged because they contain the same variable. And to collect two quantities Ghebretin or more similar use property of the distribution of multiplication over addition, for example.
2x + 3 x + 4 h is (2 + 3 + 4) Q 9 or Q, but we can not express the sum of two quantities is similar with a single. For example, the sum of A and B written A + B. And to collect 3a, 4 b 0.6 a and b use his replacement and assembly of the collection process. It is clear that these special Tsaaadanna to collect any series of the border, written in any order. And the compilation of similar border, we find that:
3a +6 a = 9 a and 4 b + b b = 5.
So 3a +4 b + 6 a + b = 9 a + 5 b.
The solution could be organized as follows:
And to collect similar amounts of non-negative or positive, we were using a private distribution of multiplication over addition. To make it clear that use the collection:
(2a – b ² c + d 6 b ² + 2 d ) and
4 (a + 3 b ² c – 4 d b ² – 3 d ) and
3 (a + 2 b ² c + d 2 b ² – 4 d ) and
(-2 A – 8 b ² c + d 6 b ² + 6 d ).
And the number 3, which appears in the border such as 2 a means that a variable multiplied by itself three times. See: the cube. Before the process of collecting such amounts arrange the border in the columns.
Algebra equations
Algebra equation include letters represent unknown numbers.
It is one of the main branches of algebra in mathematics, where the mastery of mathematics depends on a proper understanding of algebra. And uses the engineers and scientists algebra every day, and counts commercial and industrial projects on the algebra to solve many of the dilemmas faced by them. Given the importance of algebra in modern life, it is taught in schools and universities all over the world.
Symbolizes the number of anonymous letters in algebra, such as X or Y. In some of the issues can be replaced only one number is indicated. As an example note that even a simple sentence becomes + 3 = 8 should be correct to compensate for x number 5 because 5 + 3 = 8.
In some other issues, it can compensate for the code number or more. For example, in order to achieve the health of sentence constraint x + y = 12 may put Q equals 6 and Y equals 6, or Q equal to 4, and Y equal to 8. In such sentences arrest, you can get several values ​​for x makes true if the sentences given for r different values.
And admire many of the students of his ability and usefulness of algebra big, as using algebra, one can solve many of the issues that can not be resolved by using the only account. For example, say the plane cut a distance of 1710 km in four hours if the flight in the direction of the wind blowing, but cut 1370 km in five hours if the flight was blowing the opposite direction of the wind. Using algebra, we can find the speed of the plane and wind speed.
Terminology used in algebra
Exponent of the number placed on the number or variable from the left to indicate the number of times where it is used as a factor.
Signals the assembly , brackets []. And are used in algebra formulas to account for arrest.
Square or second-degree variable multiplied by the same user as any ¸ twice •.
Binomial term in algebra consists of two double-edged symbol + or the symbol -.
The number of fixed or variable scope set of one item.
Roots of the equation numbers that make the equation correct a report when you replace the variables in the equation.
Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors. Moving from Arithmetic to Algebra will look like this: Arithmetic: 3 + 4 = 3 + 4 in Algebra it would look like: x + y = y + x
The name ‘algebra’ is derived from the treatise written by the Persian mathematician 
Muhammad bin MÅ«sā al-KhwārizmÄ« titled (in Arabic “Al-Kitab al-Jabr wa-l-Muqabala” 
The development of algebra is outlined in these notes under the following headings: Egyptian algebra, Babylonian algebra, Greek geometric algebra, Diophantine algebra, Hindu algebra, Arabic algebra, European algebra since 1500, and modern algebra.
Since algebra grows out of arithmetic, recognition of new numbers – irrationals, zero, negative numbers, and complex numbers – is an important part of its history.
And later became known to science in general mathematical equations
Best way to learn and teach algebra
As you already know, algebra is an essential subject. It’s the gateway to mathematics. It’s used extensively in the sciences. And it’s an important skill in many careers.
Yet for many people Algebra is a nightmare. It causes more stress, homework tears and plain confusion than any other subject on the curriculum.
Well the good news is you don’t have to struggle with Algebra for a minute longer. Because now there’s a solution that explains Algebra in a way that anyone can quickly understand. about “How to learn algebra the easy way”. Algebra is not that difficult as everyone thinks. With some practice & hard work anyone can master it.
How to learn algebra easy way
The learning of any subject needs to understand well, and algebra is not exception to other branches of maths , as we know the maths is first thing we learn before anything else even before we go to school ,therefore it is easier than other subjects in my opinion . And started counting fingers even when you buying sweet .every one of us has knowledge of some collections like books ,cars, and so on ,it is good to use as groups we know as rats, cow ,pen . Some student surprise if you say 5x+4=24 but it will be easy to say 5cars=20£ how much the price of one car?
When we use variable numbers and letters instead of numbers only it is algebra, truly it is very fun and easy if we make more effort with understanding.
To understand it needs to make more practice and follow up the rules, addition ,subtraction, multiplication, division and equality of equations because changing sign from side to side is very important and algebra is not exception to other branch of maths .
Understanding and practice whenever you make more practice sure you will be mathematician person ,it is not difficult as many people afraid or think
1
LEARN ALGEBRA THE EASY WAY :
The key to learn and understand Mathematics is to “practice” and Algebra is no exception. Understanding the concepts is very vital, without which you are going to have difficulty learning algebra. Algebra helps in problem solving, reasoning, decision making, and applying solid strategies which is important in your day to day life especially in a job atmosphere. Consider Algebra to be a game and you would find how easy it is, you’ll see the miracle !
2
There are several techniques that can be followed to learn Algebra the easy way. Learning algebra from the textbook can be boring. Though textbooks are necessary it doesn’t always address the need for a conceptual approach. There are certain techniques that can be used to learn algebra the fun and easy way. Listed below are some of the techniques that can be used. Do some online research and you will be surprised to find a whole bunch of websites that offer a variety of fun learning methods which makes learning algebra a pleasant experience and not a nightmare. But the key is to take your time in doing a thorough research before you choose the method that is best for you, or you can do a combination of different methods if you are a person who looks for variety to boost your interest.
3
1. ANIMATED ALGEBRA : You can learn the basic principles of algebra through this method. Animation method teaches the students the concepts by helping them integrate both teaching methods. When the lessons are animated you actually learn more !
2. ALGEBRA QUIZZES : You can use softwares and learn at your own pace & best of all you don’t need a tutor to use it. What you really need is something that can help you with your own homework, not problems it already has programmed into it that barely look like what your teacher or professor was trying to explain. You can enter in your own algebra problems, and it works with you to solve them faster & make them easier to understand.
3. INTERACTIVE ALGEBRA : There are several Interactive Algebra plugins that allows the user to explore Algebra by changing variables and see what happens. This promotes an understanding of how you arrive at answers. There are websites that provide online algebra help and worksheets. They also provide interactive online games and practice problems and provide the algebra help needed.
It is difficult to recommend better methods for studying and for learning because the best methods vary from person to person. Instead, I have provided several ideas which can be the foundation to a good study program. If you just remember all the rules and procedures without truly understanding the concepts, you will no doubt have difficulty learning algebra. So the magic word is “concept”. The above techniques can help you in learning the concepts without pain in a fun environment
Read more: How to learn algebra the easy way ! | eHow.com http://www.ehow.com/how_4452787_learn-algebra-easy-way.html#ixzz1M8en5qcH
BIBLOGRAPHY
 

Linear algebra

1. INTRODUCTION:
Linear algebra comprises of the theory and application of linear system of equation, linear transformation and Eigen value problem. In linear algebra, we make a systematic use of matrix and lesser extent determinants and their properties.
Determinants were first introduced for solving linear systems and have important engineering applications in system of differential equations, electrical networks, Eigen value problems and so on. Many complicated expression occurring in electrical and mechanical systems can be elegantly simplified by expressing them in the form of determinants.
Cayley discovered matrix in the year 1860. But it was not until the twentieth century was well advanced that engineer heard of them. These days, however,
Matrices have been found to be of great utility in many branches of applied mathematics such as algebraic and differential equation, mechanics, theory of electrical circuit,
Nuclear physics, aerodynamics and astronomy with the advent of computers, the usage of matrix method has been greatly facilitated.
2. MATRIX:-
A system of m n numbers arranged in a rectangular formation along m rows and n columns an bounded by a bracket [ ] is called an m by n matrix; which is written as m*n matrix. A matrix is also denoted by a single capital letter.

(mathematics) a rectangular array of quantities or expressions set out by rows and columns; treated as a single element and manipulated according …
(geology) amass of fine-grained rock in which fossils, crystals, or gems are embedded
an enclosure within which something originates or develops (from the Latin for womb)
the body substance in which tissue cells are embedded
the formative tissue at the base of a nail

To locate any particular element of a matrix, the element respectively specify the rows and the columns. Thus aij is the
In this notation, the matrix is denoted by [aij].
3. HISTORY:-
The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway.
It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:-

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The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:-
Now the author does something quite remarkable. He sets up the coefficients of the system of three linear equations in three unknowns as a table on a ‘counting board’.

4. Other historical usages of the word “matrix” in mathematics
The word has been used in unusual ways by at least two authors of historical importance.
Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910-1913) use the word matrix in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function will be identical to its extension:
“Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e. by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.
For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g. y, by “considering” the function for all possible values of “individuals” ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e. ∀ai: Φ(ai, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” bi substituted in place of variable y:
∀bj∀ai: Φ(ai, bj)
Alfred Tarski in his 1946 Introduction to Logic used the word “matrix” synonymously with the notion of truth table as used in mathematical logic.
(5): TYPES OF MATRIX:
(A): ROW MATRIX:
A matrix having a single row is called row matrix.
e.g, A = [1234]
(B): COLUMN MATRIX:
A matrix having a single column is called column matrix.
e.g
(C): SQUARE MATRIX:
A matrix having n rows and n columns is called square matrix.
e.g. 1 2
3 4
(D): DIAGONAL MATRIX:
A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix.
e.g. 1 0
0 2
(E): UNIT MATRIX:
A diagonal matrix of order n which for all its diagonal elements, is called a unit matrix or an identity matrix of order n.
e.g. 1 0
0 1
(F): NULL MATRIX:
I all the elements of a matrix are zero; it is called a null matrix.
e.g. 0 0
0 0
6. Eigen Values?
Many problems in mathematics and physics reduce “eigenvalue problems”, and people often expend lots of effort trying to determine the eigenvalues of a general NxN system, and this is sometimes reduced to finding the roots of the Nth degree characteristic polynomial. The task of efficiently finding all the complex roots of a high-degree polynomial is non-trivial, and is complicated by the need to consider various special cases such as repeated roots.
However, depending on what we’re really trying to accomplish, we may not need to find the eigenvalues at all. Of course, the description of an action is usually simplest when its components are resolved along the directions of the eigenvectors, e.g., the eigenvectors of a rotation matrix define its axis of rotation. As a result, it’s easy to write equations for the positions of points on a rotating sphere if we work in coordinates that are aligned with the eigenvectors of the rotation, whereas if we work in some other arbitrary coordinates the equations describing the motion will be more complicated.
However, there are many situations in which the characteristic axes of the phenomenon can be exploited without ever explicitly determining the eigenvalues. For example, suppose we have the system of three continuous variables x(t), y(t), and z(t) that satisfy the equations
ax + by + cz = x’
dx + ey + fz = y’
gx + hy + iz = z’
6. Matrix eigenvalues
Eigenvalues are a special set of scalars associated with alinear system of equations(i.e., amatrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p.144).
The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent tomatrix diagonalizationand arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-calledeigenvector(or, in general, a correspondingright eigenvectorand a correspondingleft eigenvector; there is no analogous distinction between left and right for eigenvalues).
Let be alinear transformationrepresented by amatrix. If there is avectorsuch that
for somescalar, then is called the eigenvalue of with corresponding (right)eigenvector
.
Letting be asquare matrix
with eigenvalue, then the correspondingeigenvectorssatisfy
which is equivalent to the homogeneous system
Equation (4) can be written compactly as
where is theidentity matrix. As shown inCramer’s rule, alinear system of equationshas nontrivial solutionsiffthedeterminantvanishes, so the solutions of equation (5) are given by
This equation is known as thecharacteristic equationof , and the left-hand side is known as thecharacteristic polynomial.
7. Existence and multiplicity of eigenvalues
For transformations on real vector spaces, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may include complex numbers with a non-zero imaginary component. For example, a matrix representing a planar rotation of 45 degrees will not leave any non-zero vector pointing in the same direction. Over a complex vector space, the fundamental theorem of algebra guarantees that the characteristic polynomial has at least one root, and thus the linear transformation has at least one eigenvalue.
As well as distinct roots, the characteristic equation may also have repeated roots. However, having repeated roots does not imply there are multiple distinct (i.e., linearly independent) eigenvectors with that eigenvalue. The algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue.
Over a complex space, the sum of the algebraic multiplicities will equal the dimension of the vector space, but the sum of the geometric multiplicities may be smaller. In a sense, then it is possible that there may not be sufficient eigenvectors to span the entire space. This is intimately related to the question of whether a given matrix may be diagonalized by a suitable choice of coordinates.
Shear
Horizontal shear. The shear angle φ is given by k = cot φ.
Shear in the plane is a transformation in which all points along a given line remain fixed which while other points are shifted parallel to that line by a distance proportional to their perpendicular distance from the line. Shearing a plane figure does not change its area. Shear can be horizontal − along the X axis, or vertical − along the Y axis. In horizontal shear (see figure), a point P of the plane moves parallel to the X axis to the place P’ so that its coordinate y does not change while the x coordinate increments to become x’ = x + k y, where k is called the shear factor.
The matrix of a horizontal shear transformation is . The characteristic equation is λ2 − 2 λ + 1 = (1 − λ)2 = 0 which has a single, repeated root λ = 1. Therefore, the eigenvalue λ = 1 has algebraic multiplicity 2. The eigenvector(s) are found as solutions of
The last equation is equivalent to y = 0, which is a straight line along the x axis. This line represents the one-dimensional eigenspace. In the case of shear the algebraic multiplicity of the eigenvalue (2) is greater than its geometric multiplicity (1, the dimension of the eigenspace). The eigenvector is a vector along the x axis. The case of vertical shear with transformation matrix is dealt with in a similar way; the eigenvector in vertical shear is along the y axis. Applying repeatedly the shear transformation changes the direction of any vector in the plane closer and closer to the direction of the eigenvecto
Eigenvalues of 3×3 matrices
If then thecharacteristic polynomialof A is
Alternatively the characteristic polynomial of a 3×3 matrix can be written in terms of thetracetr(A)anddeterminantdet(A)as
whereI3is the 3×3identity matrix.
8. Identifying eigenvectors
With the eigenvalues in hand, we can solve sets of simultaneous linear equations to determine the corresponding eigenvectors. Since we are solving for the system, ifλ = 2then,
Now, reducing torow echelon form:
allows us to solve easily for the eigenspaceE2:
We can confirm that a simple example vector chosen from eigenspaceE2is a valid eigenvector with eigenvalueλ = 2:
Note that we can determine the degrees of freedom of the solution by the number of pivots.
IfAis arealmatrix, the characteristic polynomial will have real coefficients, but its roots will not necessarily all be real. Thecomplexeigenvalues come in pairs which areconjugates. For a real matrix, the eigenvectors of a non-real eigenvaluez, which are the solutions of(A−zI)v= 0, cannot be real.
Thespectral theoremfor symmetric matrices states that ifAis a real symmetricn-by-nmatrix, then all its eigenvalues are real, and there existnlinearly independent eigenvectors forAwhich are mutuallyorthogonal. Symmetric matrices are commonly encountered in engineering.
9: PROPERTIES OF EIGEN VALUE
(i) 1.Any square matrix A and its transpose A’ has the same eigen value
(ii) 2.The eigen value of a triangular matrix are just the diagonal element of the matrix
(iii) 3.The eigen value of an idempotent matrix are either zero or unity
(iv) 4.The sum of the eigen value of matrix is the sum of the of the element of the principal diagonal
(v) 5.The product of the eigen value of a tmatrix A is equal to its determinant
(vi) 6.If λ is the eigen value of a matrix A then 1/λ is the eigen value of A-1.
(vii) 7.If λ is the eigen value of an orthogonal matrix then 1/λ is its eigen value.
(viii) 8.If λ1, λ2,…………, λn are the eigen value of a matrix A,then Am has the eigen value λ1m λ2m……………..λnm
(ix)
(x) A is real Eigen value are real or complex conjugates in pairs.
2 -1 -1
A = 0 3 -1
0 0 2
The characteristic equation
A-λI =0
2-λ -1 -1
0 3-λ -1 = 0
0 0 2-λ
(2-λ)(3-λ)(2-λ)=0
λ =2, 2, 3
Eigen value=2, 2, 3
We can write Eigen value is the form of complex
First complex Eigen value=2+0i
Second complex Eigen value=2-0i
Hence it is prove that A is real Eigen value are real or complex conjugates in pairs.
(xi) A-1 exist if and only if 0 is not an Eigen
Value of A it has Eigen values
1/λ1,1/λ2,………………..1/λn
If λ is an Eigen value of matrix A, then 1/λ is the Eigen value of A-1
If x be the Eigen vector corresponding to λ,
Then AX=λX
Multiply by A both side,
A-1 A=A-1 λX
IX=λ (A-1 X)
X=λ (A-1X)
A-1 X=1/λ X
Shows that 1/λ is an Eigen value of the inverse matrix A-1.
If Eigen value of matrix A is 0 then,
Eigen value of inverse matrix A-1 will not exist because
Eigen value of matrix A is λ and Eigen value of inverse matrix A-1 is 1/λ.
PROFF OF THE GIVEN STATEMENT:
A=
So its eigen value are 1, 3, -2
Now,
A-1=
R2→R2+R3
R2/3
R3/-3 :
R1→R1-2R2+R3 :
So eigen vector of A-1 are 1, 3/2 -1/2.
λ 1 =1
λ 2=3
λ 3=-2
λ 1’= 1/ λ 1=1/1
λ 2’=1/ λ 2=1/3
λ 3’=1/ λ3=1/2
Its eigen value are real and also A is real and satisfied the first condition of the topic
When A-1 exist it has the eigen values 1, 1/3, -1/2 i.e. 1/λ1, 1/ λ2, 1/ λ3 satisfied the second condition of the topic.
CONLUSION:
— This is to conclude that while doing this topic “EIGN VALUE & VECTOR” I came to know many more things about EIGEN VALUE & VECTOR and their applications. which I not even heard of.It was great experience doing this topic. It has also increased my knowledge .
 

What Is Algebra?

John Pell
John Pell (1611-1685) was born in Southwick, Sussex, England, and died in Westminster, London, England.
Pell’s work was mostly based on number theory and algebra.
Pell published many books on mathematics such as Idea of Mathematicsin 1638 and the two page ‘A Refutation of Longomontanus’s Pretended Quadrature of the Circle’ in 1644.
Reverend John Wallis
John Wallis (1616-1703) was born in Ashford, Kent, England and died in Oxford, England.
In 1656, Wallis published his most famous book Arithmetica Infinitorum in which he introduced the formula /2 = (2.2.4.4.6.6.8.8.10…)/ (1.3.3.5.5.7.7.9.9…).
In another of his works, ‘Treatise on Algebra’, Wallis gives a wealth of information on algebra.
John Herschel
John Frederick William Herschel (1792-1871) was born in Slough, England and died in Kent, England.
He was a great astronomer who discovered Uranus.
In 1822, he published his first work on astronomy, a small work to calculate the eclipses of the moon.
In 1824, he published his first major work on double stars in the Transactions of the Royal Society.
Charles Babbage
Charles Babbage (1791 -1871) was born in London, England and died in London, England.
In 1821, Babbage made the Difference engine to compile tables of mathematics.
In 1856, he invented Analytical Engine, which is a general symbol manipulator and similar to today’s computers.
Sir Isaac Newton
Sir Isaac Newton (1643-1727) was born in Lincolnshire, England and died in London, England.
He was a great physicist, mathematician, and one of the greatest scientific intellects of all time.
In 1672, he published his first work on light and color in the Philosophical Transactions of the Royal Society.
In 1704, Newton’s works on pure mathematics was published and in 1707, his Cambridge lectures from 1673 to 1683 were published. ( http://www.barcodesinc.com/articles/algebra-history.htm)
How is Algebra used in daily life?
Every day in our life and all over the world we use Algebra many places as well as finances, engineering, schools, and universities we can’t do most scopes without maths.( It is actually quite common for an average person to perform simple Algebra without realizing it. For example, if you go to the grocery store and have ten dollars to spend on two dollar candy bars. This gives us the equation 2x = 10 where x is the number of candy bars you can buy. Many people don’t realize that this sort of calculation is Algebra; they just do it). (http://wiki.answers.com and http://wiki.answers.com)
Other Definitions
Algebra is the parts of mathematics where numbers and letters are used like A B or X and Y, or other symbols are used to represent unknown or variable numbers.
For examples : inA +5 = 9, A is unknown, but we can solve by subtracting 5 to both sides of the equal sign (=), like this: A+5 = 9
A+ 5 – 5 = 9 – 5
A +0 = 4
A = 4
3b+12=15 subtract both sides 12
3b+12-12=15-12
3b=3 divide both sides 3 to get the value of b which is 1 and so on
5x/5x=1 if you substitute x any number not zero the equation will be true
(Algebra is branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc. Moving from Arithmetic to Algebra will look something like this: Arithmetic: 3 + 4 = 3 + 4 in Algebra it would look like: x + y = y + )
artical http://math.about.com/cs/algebra/g/algebradef.htm
Terminology used in algebra
to make algebra easy or any other branches of maths, we must understand well all basic sign in all operations and use it right way, these signs are , subtractions ,division, addition ,multiplication. variable is also called an unknown and can be represented by letters from the alphabet letters. Operations in algebra are the same as in arithmetic: addition, subtraction, multiplication and division. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions. (Polynomials are often written in descending order, in which the terms with the largest powers are written first (like 9×2- 3x + 6). If they are written with the smallest terms appearing first, this is ascending order (like 6 – 3x + 9×2).
equation- An equation is a mathematical statement that contains an equal sign, like ax + b = c.
exponent- An exponent is a power that a number is raised to. For example, in 23, the exponent is 3.
expression- An algebraic expression consists of one or more variables, constants, and operations, like 3x-4. Each part of an expression that is added or subtracted is called atermFor example, the expression 4×2-2x+7 has three terms.
factor- The factor of a number is a number that divides that number exactly. For example, the factors of 6 are 1, 2, 3 and 6.
formula- A formula shows a mathematical relationship between expressions.
fraction- A fraction is a part of a whole, like a half, a third, a quarter, etc. For example, half of an apple is a fraction of an apple. The top number in a fraction is called the numerator; the bottom number in a fraction is called the denominator.
inequality- An inequality is a mathematical expression that contains an inequality symbol. The inequality symbols are :
> greater than (2>1)
≤ less than or equal to
≥ greater than or equal to
≠ not equal to (1≠2).
integer- The integers are the numbers …, -3, -2, -1, 0, 1, 2, ….
inverse (addition)- The inverse property of addition states that for every number a, a + (-a) = 0 (zero).
inverse (multiplication)- The inverse property of multiplication states that for every non-zero number a, a times (1/a) = 1.
matrix-
nth-
operation- An operation is a rule for taking one or two numbers as inputs and producing a number as an output. Some arithmetic operations are multiplication, division, addition, and subtraction.
polynomial- A polynomial is a sum or difference of terms; each term is:
* a constant (for example, 5)
* a constant times a variable (for example, 3x)
* a constant times the variable to a positive integer power (for example, 2×2)
* a constant times the product of variables to positive integer powers (for example, 2x3y).
monomial is a polynomial with only one term. A binomial is a polynomial that has two terms. A trinomial is a polynomial with three terms.
prime number- A prime number is a positive number that has exactly two factors, 1 and itself. Alternatively, you can think of a prime number as a number greater than one that is not the product of smaller numbers. For example, 13 is a prime number because it can only be divided evenly by 1 and 13. For another example, 14 is not a prime number because it can be divided evenly by 1, 2, 7, and 14. The number one is not a prime number because it has only one factor, 1 itself.
quadratic equation- A quadratic equation is an equation that has a second-degree term and no higher terms. A second-degree term is a variable raised to the second power, like x2, or the product of exactly two variables, like x and y.
When you graph a quadratic equation in one variable, like y = ax2+ bx + c, you get a parabola, and the solutions to the quadratic equation represent the points where the parabola crosses the x-axis.
quadratic formula- The quadratic formula is a formula that gives you a solution to the quadratic equation ax2+ bx + c = 0. The quadratic formula is obtained by solving the general quadratic equation.
radical- A radical is a symbol √ that is used to indicate the square root or nthroot of a number.
root- An nthroot of a number is a number that, when multiplied by itself n times, results in that number. For example, the number 4 is a square root of 16 because 4 x 4 equals 16. The number 2 is a cube root of 8 because 2 x 2 x 2 equals 8.
solve- When you solve an equation or a problem, you find solutions for it.
square root- The square roots of a number n are the numbers s such that s2=n. For example, the square roots of 4 are 2 and -2; the square roots of 9 are 3 and -3.
symbol- A symbol is a mark or sign that stands for something else. For example, the symbol÷meansdivide.
system of equations- A system of equations is two or more independent equations that are solved together. For example, the system of equations: x + y = 3 and x – y = 1 has a solution of x=2 and y=1.
terms- In an expression or equation, terms are numbers, variables, or numbers with variables. For example, the expression 3x has one term, the expression 4×2+ 7 has two terms.
variable- A variable is an unknown or placeholder in an algebraic expression. For example, in the expression 2x+y, x and y are variables.

+,-

(www.EnchantedLearning.com)

Learn algebra
Symbolizes the number in the account to a group that contains that number of things, for example, No. 5, always stands for a set containing 5 things.In algebra, the symbols may be replaced by numbers, but it is possible to solve the number one or more replace one icon.To learn algebra, we must first learn how to use symbols replace the numbers.And then how to create a constraint for strings of numbers.

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Groups and variables.There is a relationship between the symbols in algebra and groups of numbers.It is certain that each of us has some knowledge of groups of objects, such as collections of books, collections of postage stamps, and groups of dishes.And groups of numbers are not different for these groups a lot.One way to describe sets of numbers in algebra is that we are using one of the alphabet, such as the name of her p..Then half of the numbers of this group Bhzaretha brackets of the form {}.For example, can be expressed set of numbers from 1 to 9 as follows:
A = {1, 2.3, 4, 5.6, 7, 8.9}.
The group of odd numbers under 20 are:
B = {1.3, 5, 7.9, 11, 13.15, 17, 19}.
These examples demonstrated the models of the groups used in algebra.
Suppose that the age of four persons were respectively: 12, 15.20, 24.
Then can be written in this age group numbers.
A = {12.15, 20, 24}.
How is the age of each of them after three years?
One way to answer this question is that we write 12 +3.15 +3.20 +3 and 24 + 3.We note that the number 3 is repeated in each of the formulas ¸ • four.In algebra we can express all previous versions form a single task is m + 3 where m is any number of numbers of a group.That is, it can replace any of the numbers 12, 15, 20 or 24 m are indicated.Is called the symbol m variable, called the group a field of this variable, but No. 3 in the formula m+3 is called hard because its value is always one.Known variable in algebra as a symbol can be compensated for the number of one or more belongs to a group.
We can replace any names lead to correct reports or reports the wrong variable.For example:
Hungary is bordered by the State of the Black Sea
Report of the wrong, as in fact can not be like this report is correct only if compensated by the variable r one of the States: Bulgaria or Romania, or Turkey.The report shall be ¸ Turkey is a country bordered by the Black Sea
for example, the right one called the compensation that makes the report and called the right roots group consisting of all roots with a solution.The solution set is the previous example. {Bulgaria, Romania, Turkey}.And in reparation for not use the names to compensate for variables, but we use the numbers.
Equations known as the camel sports is equal to reflect the two formats. Phrase:
Q +7 = 12
For example, an easy equation ¸ mean the sum of the number 7 with the number equal to 12
• To solve this equation, we can do to compensate for different numbers of Q until we get a report of the equation makes the right one.If we substitute for x the equation becomes number five report is correct, and if we substitute for x any number of other reports, the equation becomes wrong.So to solve this equation set is {5}. This group contains only one root.
It is possible that the equation more than one root:
X ² + 18 = 9 o.
No. 2 highest first variable x means that the number of representative variable Q is the number of box, that number multiplied by itself once.See: box.In this equation, we can make up for X number 3:
3 Ã- 3 + 18 = 9 Ã- 3
9 + 18 = 27
27 = 27
We can also compensate for X number 6:
6 Ã- 6 + 18 = 9 Ã- 6
36 + 18 = 54
54 = 54
Any other compensation for making the equation Q report wrong.Then 3 and 6 are the root of the equation.Thus, the solution set is {3.6}.
There are also equations having no roots:
X = + 3
If we substitute for x any number, this equation becomes a false report, and a solution is called the group of free and symbolized by the symbol {}.
and some of the equations, an infinite number (for high standards) from the roots.
(X + 1) ² = x ² + 2 x +1
In this equation if we substitute for x any number we get the right report, the Group resolved to contain all the numbers http://nabad-alkloop.com/vb/showthread.php?t=38762
What is best way to learn and teach algebra?
Step-by-step equations solving is the key of teaching and learning. To find fully worked-out answers and learn how to solve math problems, one step at a time. Studying worked-out solutions is a proven method to help you retain information. Don’t just look for the answer in the back of the book;
There are five laws basic principles of math governing operations: multiplication addition subtract and expressing the variables and can be compensated for any number
Algebra is anessential subject. It’s the gateway to mathematics. It’s used extensively in the sciences. And it’s an important skill in many careers. Many people think, it is a nightmare and causes more stress, homework tears and plain confusion than any other subject on the curriculum but that is not true.
Theimportance of understanding equation
Connotation and denotation on extension of a concept two opposite yet complementary aspects is clarified the extension is defined vice versa understanding the concept equation includes its connotation and denotations.
This session of observed lessons will show the essential nature or the equation is consolidated by designing problem variation putting emphasis on clarifying the connotation and differentiation the boundary of the set of object in the extension. (Page 559 Jifa cai)
What’s the best formula for teaching algebra?
Immersing students in their course work, or easing them into learning the new skills or does a combination of the two techniques adds up to the best strategy? Researchers at the Centre for Social Organization of Schools at Johns Hopkins are aiming to find out through a federally funded study that will span 18 schools in five states this fall.
The study, now in its second year of data collection, will evaluate two ways to teach algebra to ninth-graders, determining if one approach is more effective in increasing mathematics skills and performance or whether the two approaches are equally effective. Participating schools in North Carolina, Florida, Ohio, Utah and Virginia will be randomly assigned to one of two strategies for the 2009-2010 school year; to be eligible, students must not have previously taken Algebra I. Twenty-eight high schools were studied during the 2008-2009 school year.
One strategy, called Stretch Algebra, is a yearlong course in Algebra 1 with students attending classes of 70 to 90 minutes a day for two semesters. This approach gives students a “double dose” of algebra, with time to work on fundamental mathematics skills as needed.
The second strategy is a sequence of two courses, also taught in extended class periods. During the first semester, students take a course called Transition to Advanced Mathematics, followed by the district’s Algebra I course in the second semester. The first-semester course was developed by researchers and curriculum writers at Johns Hopkins to fill gaps in fundamental skills, develop mathematics reasoning and build students’ confidence in their abilities.
“The question is, Is it better for kids to get into algebra and do algebra, or to give kids the extra time so the teacher can concentrate more on concepts started in middle schools?” said Ruth Curran Neild, a research scientist at Johns Hopkins and one of the study’s principal investigators.
Teachers using both strategies will receive professional development. Mathematics coaches will provide weekly support to those who are teaching the two-course approach; the study will provide teacher guides and hands-on materials for students in Transition to Advanced Mathematics. Johns Hopkins researchers will be collecting data throughout the school year. Findings are expected during the 2010-2011 school year. http://gazette.jhu.edu/2009/08/17/calculating-the-best-way-for-teaching-algebra/
Learn Algebra, the easy way
The key to learn and understand Mathematics is to “practice more and more” and Algebra is no exception. Understanding the concepts is very vital. There are several techniques that can be followed to learn Algebra the easy way. Learning algebra from the textbook can be boring. Though textbooks are necessary it doesn’t always address the need for a conceptual approach. There are certain techniques that can be used to learn algebra the fun and easy way. Listed below are some of the techniques that can be used. Do some online research and you will be surprised to find a whole bunch of websites that offer a variety of fun learning methods which makes learning algebra a pleasant experience and not a nightmare. But the key is to take your time in doing a thorough research before you choose the method that is best for you, or you can do a combination of different methods if you are a person who looks for variety to boost your interest.
1. ANIMATED ALGEBRA : You can learn the basic principles of algebra through this method. Animation method teaches the students the concepts by helping them integrate both teaching methods. When the lessons are animated you actually learn more !
2. ALGEBRA QUIZZES : You can use software’s and learn at your own pace & best of all you don’t need a tutor to use it. What you really need is something that can help you with your own homework, not problems it already has programmed into it that barely look like what your teacher or professor was trying to explain. You can enter in your own algebra problems, and it works with you to solve them faster & make them easier to understand.
3. INTERACTIVE ALGEBRA : There are several Interactive Algebra plugins that allows the user toexploreAlgebra by changing variables and see what happens. This promotes an understanding of how you arrive at answers. There are websites that provide online algebra help and worksheets. They also provide interactive onlinegamesand practice problems and provide the algebra help needed.
It is difficult to recommend better methods for studying and for learning because the best methods vary from person to person. Instead, I have provided several ideas which can be the foundation to a good study program. If you just remember all the rules and procedures without truly understanding the concepts, you will have difficulty learning algebra. (http://www.ehow.com/how_4452787_learn-algebra-easy-way.html)
 

The History of Algebra

John Pell
John Pell (1611-1685) was born in Southwick, Sussex, England, and died in Westminster, London, England.
Pell’s work was mostly based on number theory and algebra.
Pell published many books on mathematics such as Idea of Mathematics in 1638 and the two page ‘A Refutation of Longomontanus’s Pretended Quadrature of the Circle’ in 1644.
Reverend John Wallis
John Wallis (1616-1703) was born in Ashford, Kent, England and died in Oxford, England.
In 1656, Wallis published his most famous book Arithmetica Infinitorum in which he introduced the formula /2 = (2.2.4.4.6.6.8.8.10…)/ (1.3.3.5.5.7.7.9.9…).
In another of his works, ‘Treatise on Algebra’, Wallis gives a wealth of information on algebra.
John Herschel
John Frederick William Herschel (1792-1871) was born in Slough, England and died in Kent, England.
He was a great astronomer who discovered Uranus.
In 1822, he published his first work on astronomy, a small work to calculate the eclipses of the moon.
In 1824, he published his first major work on double stars in the Transactions of the Royal Society.
Charles Babbage
Charles Babbage (1791 -1871) was born in London, England and died in London, England.
In 1821, Babbage made the Difference engine to compile tables of mathematics.
In 1856, he invented Analytical Engine, which is a general symbol manipulator and similar to today’s computers.
Sir Isaac Newton
Sir Isaac Newton (1643-1727) was born in Lincolnshire, England and died in London, England.
He was a great physicist, mathematician, and one of the greatest scientific intellects of all time.
In 1672, he published his first work on light and color in the Philosophical Transactions of the Royal Society.
In 1704, Newton’s works on pure mathematics was published and in 1707, his Cambridge lectures from 1673 to 1683 were published. ( http://www.barcodesinc.com/articles/algebra-history.htm)
How is Algebra used in daily life?
Every day in our life and all over the world we use Algebra many places as well as finances, engineering, schools, and universities we can’t do most scopes without maths.( It is actually quite common for an average person to perform simple Algebra without realizing it. For example, if you go to the grocery store and have ten dollars to spend on two dollar candy bars. This gives us the equation 2x = 10 where x is the number of candy bars you can buy. Many people don’t realize that this sort of calculation is Algebra; they just do it). (http://wiki.answers.com and http://wiki.answers.com)
Other Definitions
Algebra is the parts of mathematics where numbers and letters are used like A B or X and Y, or other symbols are used to represent unknown or variable numbers.
For examples : in A +5 = 9, A is unknown, but we can solve by subtracting 5 to both sides of the equal sign (=), like this: A+5 = 9
A+ 5 – 5 = 9 – 5
A +0 = 4
A = 4
3b+12=15 subtract both sides 12
3b+12-12=15-12
3b=3 divide both sides 3 to get the value of b which is 1 and so on
5x/5x=1 if you substitute x any number not zero the equation will be true
(Algebra is branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc. Moving from Arithmetic to Algebra will look something like this: Arithmetic: 3 + 4 = 3 + 4 in Algebra it would look like: x + y = y + )
artical http://math.about.com/cs/algebra/g/algebradef.htm
Terminology used in algebra
to make algebra easy or any other branches of maths, we must understand well all basic sign in all operations and use it right way, these signs are , subtractions ,division, addition ,multiplication. variable is also called an unknown and can be represented by letters from the alphabet letters. Operations in algebra are the same as in arithmetic: addition, subtraction, multiplication and division. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions. (Polynomials are often written in descending order, in which the terms with the largest powers are written first (like 9×2 – 3x + 6). If they are written with the smallest terms appearing first, this is ascending order (like 6 – 3x + 9×2).
equation – An equation is a mathematical statement that contains an equal sign, like ax + b = c.
exponent – An exponent is a power that a number is raised to. For example, in 23, the exponent is 3.
expression – An algebraic expression consists of one or more variables, constants, and operations, like 3x-4. Each part of an expression that is added or subtracted is called a term For example, the expression 4×2-2x+7 has three terms.
factor – The factor of a number is a number that divides that number exactly. For example, the factors of 6 are 1, 2, 3 and 6.
formula – A formula shows a mathematical relationship between expressions.
fraction – A fraction is a part of a whole, like a half, a third, a quarter, etc. For example, half of an apple is a fraction of an apple. The top number in a fraction is called the numerator; the bottom number in a fraction is called the denominator.
inequality – An inequality is a mathematical expression that contains an inequality symbol. The inequality symbols are :
> greater than (2>1)
≤ less than or equal to
≥ greater than or equal to
≠ not equal to (1≠2).
integer – The integers are the numbers …, -3, -2, -1, 0, 1, 2, ….
inverse (addition) – The inverse property of addition states that for every number a, a + (-a) = 0 (zero).
inverse (multiplication) – The inverse property of multiplication states that for every non-zero number a, a times (1/a) = 1.
matrix –
nth –
operation – An operation is a rule for taking one or two numbers as inputs and producing a number as an output. Some arithmetic operations are multiplication, division, addition, and subtraction.
polynomial – A polynomial is a sum or difference of terms; each term is:
a constant (for example, 5)
a constant times a variable (for example, 3x)
a constant times the variable to a positive integer power (for example, 2×2)
a constant times the product of variables to positive integer powers (for example, 2x3y).
monomial is a polynomial with only one term. A binomial is a polynomial that has two terms. A trinomial is a polynomial with three terms.
prime number – A prime number is a positive number that has exactly two factors, 1 and itself. Alternatively, you can think of a prime number as a number greater than one that is not the product of smaller numbers. For example, 13 is a prime number because it can only be divided evenly by 1 and 13. For another example, 14 is not a prime number because it can be divided evenly by 1, 2, 7, and 14. The number one is not a prime number because it has only one factor, 1 itself.
quadratic equation – A quadratic equation is an equation that has a second-degree term and no higher terms. A second-degree term is a variable raised to the second power, like x2, or the product of exactly two variables, like x and y.
When you graph a quadratic equation in one variable, like y = ax2 + bx + c, you get a parabola, and the solutions to the quadratic equation represent the points where the parabola crosses the x-axis.
quadratic formula – The quadratic formula is a formula that gives you a solution to the quadratic equation ax2 + bx + c = 0. The quadratic formula is obtained by solving the general quadratic equation.
radical – A radical is a symbol √ that is used to indicate the square root or nth root of a number.
root – An nth root of a number is a number that, when multiplied by itself n times, results in that number. For example, the number 4 is a square root of 16 because 4 x 4 equals 16. The number 2 is a cube root of 8 because 2 x 2 x 2 equals 8.
solve – When you solve an equation or a problem, you find solutions for it.
square root – The square roots of a number n are the numbers s such that s2=n. For example, the square roots of 4 are 2 and -2; the square roots of 9 are 3 and -3.
symbol – A symbol is a mark or sign that stands for something else. For example, the symbol ÷ means divide.
system of equations – A system of equations is two or more independent equations that are solved together. For example, the system of equations: x + y = 3 and x – y = 1 has a solution of x=2 and y=1.
terms – In an expression or equation, terms are numbers, variables, or numbers with variables. For example, the expression 3x has one term, the expression 4×2 + 7 has two terms.
variable – A variable is an unknown or placeholder in an algebraic expression. For example, in the expression 2x+y, x and y are variables.
+, –
(www.EnchantedLearning.com)
Learn algebra
Symbolizes the number in the account to a group that contains that number of things, for example, No. 5, always stands for a set containing 5 things. In algebra, the symbols may be replaced by numbers, but it is possible to solve the number one or more replace one icon. To learn algebra, we must first learn how to use symbols replace the numbers. And then how to create a constraint for strings of numbers.

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Groups and variables. There is a relationship between the symbols in algebra and groups of numbers. It is certain that each of us has some knowledge of groups of objects, such as collections of books, collections of postage stamps, and groups of dishes. And groups of numbers are not different for these groups a lot. One way to describe sets of numbers in algebra is that we are using one of the alphabet, such as the name of her p.. Then half of the numbers of this group Bhzaretha brackets of the form {}. For example, can be expressed set of numbers from 1 to 9 as follows:
A = {1, 2.3, 4, 5.6, 7, 8.9}.
The group of odd numbers under 20 are:
B = {1.3, 5, 7.9, 11, 13.15, 17, 19}.
These examples demonstrated the models of the groups used in algebra.
Suppose that the age of four persons were respectively: 12, 15.20, 24.
Then can be written in this age group numbers.
A = {12.15, 20, 24}.
How is the age of each of them after three years?
One way to answer this question is that we write 12 +3.15 +3.20 +3 and 24 + 3. We note that the number 3 is repeated in each of the formulas ¸ • four. In algebra we can express all previous versions form a single task is m + 3 where m is any number of numbers of a group. That is, it can replace any of the numbers 12, 15, 20 or 24 m are indicated. Is called the symbol m variable, called the group a field of this variable, but No. 3 in the formula m+3 is called hard because its value is always one. Known variable in algebra as a symbol can be compensated for the number of one or more belongs to a group.
We can replace any names lead to correct reports or reports the wrong variable. For example:
Hungary is bordered by the State of the Black Sea
• Report of the wrong, as in fact can not be like this report is correct only if compensated by the variable r one of the States: Bulgaria or Romania, or Turkey. The report shall be ¸ Turkey is a country bordered by the Black Sea
• for example, the right one called the compensation that makes the report and called the right roots group consisting of all roots with a solution. The solution set is the previous example. {Bulgaria, Romania, Turkey}. And in reparation for not use the names to compensate for variables, but we use the numbers.
Equations known as the camel sports is equal to reflect the two formats. Phrase:
Q +7 = 12
For example, an easy equation ¸ mean the sum of the number 7 with the number equal to 12
• To solve this equation, we can do to compensate for different numbers of Q until we get a report of the equation makes the right one. If we substitute for x the equation becomes number five report is correct, and if we substitute for x any number of other reports, the equation becomes wrong. So to solve this equation set is {5}. This group contains only one root.
It is possible that the equation more than one root:
X ² + 18 = 9 o.
No. 2 highest first variable x means that the number of representative variable Q is the number of box, that number multiplied by itself once. See: box. In this equation, we can make up for X number 3:
3 Ã- 3 + 18 = 9 Ã- 3
9 + 18 = 27
27 = 27
We can also compensate for X number 6:
6 Ã- 6 + 18 = 9 Ã- 6
36 + 18 = 54
54 = 54
Any other compensation for making the equation Q report wrong. Then 3 and 6 are the root of the equation. Thus, the solution set is {3.6}.
There are also equations having no roots:
X = + 3
If we substitute for x any number, this equation becomes a false report, and a solution is called the group of free and symbolized by the symbol {}.
and some of the equations, an infinite number (for high standards) from the roots.
(X + 1) ² = x ² + 2 x +1
In this equation if we substitute for x any number we get the right report, the Group resolved to contain all the numbers http://nabad-alkloop.com/vb/showthread.php?t=38762
What is best way to learn and teach algebra?
Step-by-step equations solving is the key of teaching and learning. To find fully worked-out answers and learn how to solve math problems, one step at a time. Studying worked-out solutions is a proven method to help you retain information. Don’t just look for the answer in the back of the book;
There are five laws basic principles of math governing operations: multiplication addition subtract and expressing the variables and can be compensated for any number
Algebra is an essential subject. It’s the gateway to mathematics. It’s used extensively in the sciences. And it’s an important skill in many careers. Many people think, it is a nightmare and causes more stress, homework tears and plain confusion than any other subject on the curriculum but that is not true.
The importance of understanding equation
Connotation and denotation on extension of a concept two opposite yet complementary aspects is clarified the extension is defined vice versa understanding the concept equation includes its connotation and denotations.
This session of observed lessons will show the essential nature or the equation is consolidated by designing problem variation putting emphasis on clarifying the connotation and differentiation the boundary of the set of object in the extension. (Page 559 Jifa cai)
What’s the best formula for teaching algebra?
Immersing students in their course work, or easing them into learning the new skills or does a combination of the two techniques adds up to the best strategy? Researchers at the Centre for Social Organization of Schools at Johns Hopkins are aiming to find out through a federally funded study that will span 18 schools in five states this fall.
The study, now in its second year of data collection, will evaluate two ways to teach algebra to ninth-graders, determining if one approach is more effective in increasing mathematics skills and performance or whether the two approaches are equally effective. Participating schools in North Carolina, Florida, Ohio, Utah and Virginia will be randomly assigned to one of two strategies for the 2009-2010 school year; to be eligible, students must not have previously taken Algebra I. Twenty-eight high schools were studied during the 2008-2009 school year.
One strategy, called Stretch Algebra, is a yearlong course in Algebra 1 with students attending classes of 70 to 90 minutes a day for two semesters. This approach gives students a “double dose” of algebra, with time to work on fundamental mathematics skills as needed.
The second strategy is a sequence of two courses, also taught in extended class periods. During the first semester, students take a course called Transition to Advanced Mathematics, followed by the district’s Algebra I course in the second semester. The first-semester course was developed by researchers and curriculum writers at Johns Hopkins to fill gaps in fundamental skills, develop mathematics reasoning and build students’ confidence in their abilities.
“The question is, Is it better for kids to get into algebra and do algebra, or to give kids the extra time so the teacher can concentrate more on concepts started in middle schools?” said Ruth Curran Neild, a research scientist at Johns Hopkins and one of the study’s principal investigators.
Teachers using both strategies will receive professional development. Mathematics coaches will provide weekly support to those who are teaching the two-course approach; the study will provide teacher guides and hands-on materials for students in Transition to Advanced Mathematics. Johns Hopkins researchers will be collecting data throughout the school year. Findings are expected during the 2010-2011 school year. http://gazette.jhu.edu/2009/08/17/calculating-the-best-way-for-teaching-algebra/
Learn Algebra, the easy way
The key to learn and understand Mathematics is to “practice more and more” and Algebra is no exception. Understanding the concepts is very vital. There are several techniques that can be followed to learn Algebra the easy way. Learning algebra from the textbook can be boring. Though textbooks are necessary it doesn’t always address the need for a conceptual approach. There are certain techniques that can be used to learn algebra the fun and easy way. Listed below are some of the techniques that can be used. Do some online research and you will be surprised to find a whole bunch of websites that offer a variety of fun learning methods which makes learning algebra a pleasant experience and not a nightmare. But the key is to take your time in doing a thorough research before you choose the method that is best for you, or you can do a combination of different methods if you are a person who looks for variety to boost your interest.
1. ANIMATED ALGEBRA : You can learn the basic principles of algebra through this method. Animation method teaches the students the concepts by helping them integrate both teaching methods. When the lessons are animated you actually learn more !
2. ALGEBRA QUIZZES : You can use software’s and learn at your own pace & best of all you don’t need a tutor to use it. What you really need is something that can help you with your own homework, not problems it already has programmed into it that barely look like what your teacher or professor was trying to explain. You can enter in your own algebra problems, and it works with you to solve them faster & make them easier to understand.
3. INTERACTIVE ALGEBRA : There are several Interactive Algebra plugins that allows the user to explore Algebra by changing variables and see what happens. This promotes an understanding of how you arrive at answers. There are websites that provide online algebra help and worksheets. They also provide interactive online games and practice problems and provide the algebra help needed.
It is difficult to recommend better methods for studying and for learning because the best methods vary from person to person. Instead, I have provided several ideas which can be the foundation to a good study program. If you just remember all the rules and procedures without truly understanding the concepts, you will have difficulty learning algebra. (http://www.ehow.com/how_4452787_learn-algebra-easy-way.html)
 

Hopf Algebra Project

0
Definition0.7
The wedge product is the product in an exterior algebra. If α, β are differential k-forms of degree p, g respectively, then
 α∧β=(-1)pq β∧α, is not in general commutative, but is associative,
(α∧β)∧u= α∧(β∧u) and bilinear
(c1 α1+c2 α2)∧ β= c1( α1∧ β) + c2( α2∧ β)
α∧( c1 β1+c2 β2)= c1( α∧ β1) + c2( α∧ β2).    (Becca 2014)
Chapter 1
Definition1.1
“Let (A, m, η) be an algebra over k and write mop (ab) = ab ꓯ a, bϵ A where mop=mτΑ,Α. Thus ab=ba ꓯa, b ϵA. The (A, mop, η) is the opposite algebra.”
Definition1.2
“A co-algebra C is

A vector space over K
A map Δ: C→C âŠ-  C which is coassociative in the sense of ∑ (c(1)(1) âŠ-   c(1)(2) âŠ-  c(2))= ∑ (c(1) âŠ-   c(2)(1) âŠ-  c(2)c(2) )   ꓯ cϵC (Δ called the co-product)
A map ε: C→ k obeying ∑[ε((c(1))c(2))]=c= ∑[(c(1)) εc(2))] ꓯ cϵC ( ε called the counit)”

Co-associativity and co-unit element can be expressed as commutative diagrams as follow:

Figure 1: Co-associativity map Δ

Figure 2: co-unit element map ε
Definition1.3
“A bi-algebra H is

An algebra (H, m ,η)
A co-algebra (H, Δ, ε)
Δ,ε are algebra maps, where HâŠ-  H has the tensor product algebra structure (hâŠ- g)(h‘âŠ-  g‘)= hh‘âŠ-   gg‘ ꓯh, h‘, g, g‘ ϵH. “

A representation of Hopf algebras as diagrams is the following:

Definition1.4
“A Hopf Algebra H is

A bi-algebra H, Δ, ε, m, η
A map S : H→ H such that ∑ [(Sh(1))h(2) ]= ε(h)= ∑ [h(1)Sh(2) ]ꓯ hϵH

The axioms that make a simultaneous algebra and co-algebra into Hopf algebra is
Ï„:  HâŠ- H→HâŠ-H
Is the map Ï„(hâŠ-g)=gâŠ-h called the flip map ꓯ h, g ϵ H.”
Definition1.5
“Hopf Algebra is commutative if it’s commutative as algebra. It is co-commutative if it’s co-commutative as a co-algebra, τΔ=Δ. It can be defined as S2=id.
A commutative algebra over K is an algebra (A, m, η) over k such that m=mop.”
Definition1.6
“Two Hopf algebras H,H‘ are dually paired by a map : H’ H →k if,
=ψ,Δh>, =ε(h)
g  >=, ε(φ)=
=
ꓯ φ, ψϵ H’ and h, g ϵH.
Let (C, Δ,ε) be a co-algebra over k. The co-algebra (C, Δcop, ε) is the opposite co-algebra.
A co-commutative co-algebra over k is a co-algebra (C, Δ, ε) over k such that Δ= Δcop.”
Definition1.7
“A bi-algebra or Hopf algebra H acts on algebra A (called H-module algebra) if:

H acts on A as a vector space.
The product map m: AA→A commutes with the action of H
The unit map η: k→ A commutes with the action of H.

From b,c we come to the next action
h⊳(ab)=∑(h(1)⊳a)(h(2)⊳b), h⊳1= ε(h)1, ꓯa, b ϵ A, h ϵ H
This is the left action.”
Definition1.8
“Let (A, m, η) be algebra over k and is a left H- module along with a linear map m: AâŠ-A→A and a scalar multiplication η: k âŠ- A→A if the following diagrams commute.”

Figure 3: Left Module map
Definition1.9
“Co-algebra (C, Δ, ε) is H-module co-algebra if:

C is an H-module
Δ: C→CC and ε: C→ k commutes with the action of H. (Is a right C- co-module).

Explicitly,
Δ(h⊳c)=∑h(1)⊳c(1)⨂h(2)⊳c(2), ε(h⊳c)= ε(h)ε(c), ꓯh ϵ H, c ϵ C.”

 Definition1.10
“A co-action of a co-algebra C on a vector space V is a map β: V→C⨂V such that,

(id⨂β) ∘β=(Δ⨂ id )β;
 id =(ε⨂id )∘β.”

Definition1.11
“A bi-algebra or Hopf algebra H co-acts on an algebra A (an H- co-module algebra) if:

A is an H- co-module
The co-action β: A→ H⨂A is an algebra homomorphism, where H⨂A has the tensor product algebra structure.”

Definition1.12
“Let C be co- algebra (C, Δ, ε), map β: A→ H⨂A is a right C- co- module if the following diagrams commute.”

Figure 6:Co-algebra of a right co-module
“Sub-algebras, left ideals and right ideals of algebra have dual counter-parts in co-algebras. Let (A, m, η) be algebra over k and suppose that V is a left ideal of A. Then m(A⨂V)⊆V. Thus the restriction of m to A⨂V determines a map A⨂V→V. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product Δ restricts to a map V→C⨂V.”
Definition1.13
“Let V be a subspace of a co-algebra C over k. Then V is a sub-co-algebra of C if Δ(V)⊆V⨂V, for left co-ideal Δ(V)⊆C⨂V and for right co-ideal Δ(V)⊆V⨂C.”
Definition1.14
“Let V be a subspace of a co-algebra C over k. The unique minimal sub-co-algebra of C which contains V is the sub-co-algebra of C generated by V.”
Definition1.15
“A simple co-algebra is a co-algebra which has two sub-co-algebras.”
Definition1.16
“Let C be co-algebra over k. A group-like element of C is c ϵC with satisfies, Δ(s)=s⨂s  and ε(s)=1 ꓯ s ϵS. The set of group-like elements of C is denoted G(C).”
Definition1.17
“Let S be a set. The co-algebra k[S] has a co-algebra structure determined by
Δ(s)=s⨂s  and ε(s)=1
ꓯ s ϵS. If S=∅ we set C=k[∅]=0.
Is the group-like co-algebra of S over k.”
Definition1.18
“The co-algebra C over k with basis {co, c1, c2,…..} whose co-product and co-unit is satisfy by Δ(cn)= ∑cn-l⨂cl and ε(cn)=δn,0 for l=1,….,n and for all n≥0. Is denoted by P∞(k). The sub-co-algebra which is the span of co, c1, c2,…,cn is denoted Pn(k).”
Definition1.19
“A co-matrix co-algebra over k is a co-algebra over k isomorphic to Cs(k) for some finite set S. The co-matrix identities are:

Δ(ei, j)= ∑ei, l⨂el, j
ε(ei, j)=δi, j

∀ i, j ϵS. Set C∅(k)=(0).”
Definition1.20
“Let S be a non-empty finite set. A standard basis for Cs(k) is a basis {c i ,j}I, j ϵS for Cs(k) which satisfies the co-matrix identities.”
Definition1.21
“Let (C, Δc, εc) and (D, ΔD, εD) be co-algebras over the field k. A co-algebra map f: C→D is a linear map of underlying vector spaces such that ΔD∘f=(f⨂f)∘ Δc and εD∘f= εc. An isomorphism of co-algebras is a co-algebra map which is a linear isomorphism.”
Definition1.22
“Let C be co-algebra over the field k. A co-ideal of C is a subspace I of C such that ε (I) = (0) and Δ (Ι) ⊆ I⨂C+C⨂I.”
Definition1.23
“The co-ideal Ker (ε) of a co-algebra C over k is denoted by C+.”
Definition1.24
“Let I be a co-ideal of co-algebra C over k. The unique co-algebra structure on C /I such that the projection Ï€: C→ C/I is a co-algebra map, is the quotient co-algebra structure on C/I.”
Definition1.25
“The tensor product of co-algebra has a natural co-algebra structure as the tensor product of vector space CâŠ-D is a co-algebra over k where Δ(c(1)⨂d(1))⨂( c(2)⨂d(2)) and ε(c⨂d)=ε(c)ε(d) ∀ c in C and d in D.”
Definition1.26
“Let C be co-algebra over k. A skew-primitive element of C is a cϵC which satisfies Δ(c)= g⨂c +c⨂h, where c, h ϵG(c). The set of g:h-skew primitive elements of C is denoted  by
Pg,h (C).”
Definition1.27
“Let C be co-algebra over a field k. A co-commutative element of C is cϵC such that Δ(c) = Δcop(c). The set of co-commutative elements of C is denoted by Cc(C).
Cc(C) ⊆C.”
Definition1.28
“The category whose objects are co-algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Coalg.”
Definition1.29
“The category whose objects are algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Alg.”
Definition1.30
“Let (C, Δ, ε) be co-algebra over k. The algebra (Câˆ-, m, η) where m= Δâˆ-| Câˆ-⨂Câˆ-, η (1) =ε, is the dual algebra of (C, Δ, ε).”
Definition1.31
“Let A be algebra over the field k. A locally finite A-module is an A-module M whose finitely generated sub-modules are finite-dimensional. The left and right C∖module actions on C are locally finite.”
Definition1.32
“Let A be algebra over the field k. A derivation of A is a linear endomorphism F of A such that F (ab) =F (a) b-aF(b) for all a, b ϵA.
For fixed b ϵA note that F: A→A defined by F(a)=[a, b]= ab- ba  for all a ϵA is a derivation of A.”
Definition1.33
“Let C be co-algebra over the field k. A co-derivation of C is a linear endomorphism f of C such that Δ∘f= (f⨂IC + IC ⨂f) ∘Δ.”
Definition1.34
“Let A and B ne algebra over the field k. The tensor product algebra structure on A⨂B is determined by (a⨂b)(a’⨂b’)= aa’⨂bb’ ꓯ a, a’ϵA and b, b’ϵB.”
Definition1.35
“Let X, Y be non-empty subsets of an algebra A over the field k. The centralizer of Y in X is
ZX(Y) = {xϵX|yx=xy ꓯyϵY}
For y ϵA the centralizer of y in X is ZX(y) = ZX({y}).”
Definition1.36
“The centre of an algebra A over the field Z (A) = ZA(A).”
Definition1.37
“Let (S, ≤) be a partially ordered set which is locally finite, meaning that ꓯ, I, jϵS which satisfy i≤j the interval [i, j] = {lϵS|i≤l≤j} is a finite set. Let S= {[i, j] |I, jϵS, i≤j} and let A be the algebra which is the vector space of functions f: S→k under point wise operations whose product is given by
(f⋆g)([i, j])=f([i, l])g([l, j])  i≤l≤j
For all f, g ϵA and [i, j]ϵS and whose unit is given by 1([I,j])= δi,j ꓯ[I,j]ϵS.”
Definition1.38
“The algebra of A over the k described above is the incidence algebra of the locally finite partially ordered set (S, ≤).”
Definition1.39
“Lie co-algebra over k is a pair (C, δ), where C is a vector space over k and δ: C→C⨂C is a linear map, which satisfies:
τ∘δ=0 and (Ι+(τ⨂Ι)∘(Ι⨂τ)+(Ι⨂τ)∘ (τ⨂Ι))∘(Ι⨂δ)∘δ=0
Ï„=Ï„C,C and I is the appropriate identity map.”
Definition1.40
“Suppose that C is co-algebra over the field k. The wedge product of subspaces U and V is U∧V = Δ-1(U⨂C+ C⨂V).”
Definition1.41
“Let C be co-algebra over the field k. A saturated sub-co-algebra of C is a sub-co-algebra D of C such that U∧V⊆D, ꓯ U, V of D.”
Definition1.42
“Let C be co-algebra over k and (N, ρ) be a left co-module. Then U∧X= ρ-1(U⨂N+ C⨂X) is the wedge product of subspaces U of C and X of N.”
Definition1.43
“Let C be co-algebra over k and U be a subspace of C. The unique minimal saturated sub-co-algebra of C containing U is the saturated closure of U in C.”
Definition1.44
“Let (A, m, η) be algebra over k. Then,

A∘=m∖1(Aâˆ-⨂Aâˆ- )
(A∘, Δ, ε) is a co-algebra over k, where Δ= mâˆ-| A∘ and ε=ηâˆ-.

Τhe co-algebra (A∘, Δ, ε) is the dual co-algebra of (A, m, η).
Also we denote A∘ by a∘ and Δ∘= a∘(1)⨂ a∘(2), ꓯ a∘ ϵ A∘.”
Definition1.45
“Let A be algebra over k. An η:ξ- derivation of A is a linear map f: A→k which satisfies f(ab)= η(a)f(b)+f(a) ξ(b), ꓯ a, bϵ A and η, ξ ϵ Alg(A, k).”
Definition1.46
“The full subcategory of k-Alg (respectively of k-Co-alg) whose objects are finite dimensional algebras (respectively co-algebras) over k is denoted k-Alg fd (respectively     k-Co-alg fd).”
Definition1.47
“A proper algebra over k is an algebra over k such that the intersection of the co-finite ideals of A is (0), or equivalently the algebra map jA:A→(A∘)*, be linear map defined by jA(a)(a∘)=a∘(a), a ϵA and a∘ϵA∘. Then:

jA:A→(A∘)* is an algebra map
Ker(jA) is the intersection of the co-finite ideals of A
Im(jA) is a dense subspace of (A∘)*.

Is one-to-one.”
Definition1.48
“Let A (respectively C) be an algebra (respectively co-algebra ) over k. Then A (respectively C) is reflexive if jA:A→(A∘)*, as defined before and jC:C→(C*)∘, defined as:
jC(c)(c*)=c*(c), ꓯ c*ϵC* and cϵC. Then:

Im(jC)⊆(C*)∘ and jC:C→(C*)∘ is a co-algebra map.
jC is one-to-one.
Im(jC) is the set of all aϵ(C*)* which vanish on a closed co-finite ideal of C*.

Is an isomorphism.”
Definition1.49
“Almost left noetherian algebra over k is an algebra over k whose co-finite left ideal are finitely generated. (M is called almost noetherian if every co-finite submodule of M is finitely generated).”
Definition1.50
“Let f:U→V be a map of vector spaces over k. Then f is an almost one-to-one linear map if ker(f) is finite-dimensional, f is an almost onto linear map if Im(f) is co-finite subspace of V and f is an almost isomorphism if f is an almost one-to-one and an almost linear map.”
Definition1.51
“Let A be algebra over k and C be co-algebra over k. A pairing of A and C is a bilinear map
 β: AÃ-C→k which satisfies, β(ab,c)= β (a, c(1))β (b, c(2)) and β(1, c) = ε(c), ꓯ a, b ϵ A and        c ϵC.”
Definition1.52
“Let V be a vector space over k. A co-free co-algebra on V is a pair (Ï€, Tco(V)) such that:

Tco(V) is a co-algebra over k and π: Tco(V)→T is a linear map.
If C is a co-algebra over k and f:C→V is a linear map,∃ a co-algebra map F: C→ Tco(V) determined by π∘F=f.”

Definition1.53
“Let V be a vector space over k. A co-free co-commutative co-algebra on V is any pair (Ï€, C(V)) which satisfies:

C(V) is a co-commutative co-algebra over k and π:C(V)→V is a linear map.
If C is a co-commutative co-algebra over k and f: C→V is linear map, ∃ co-algebra map F:C →C(V) determined by π∘F=f. “                  (Majid 2002, Radford David E)

Chapter 2
Proposition (Anti-homomorphism property of antipodes) 2.1
“The antipode of a Hopf algebra is unique and obey S(hg)=S(g)S(h), S(1)=1 and (S⨂S)∘Δh=τ∘Δ∘Sh, εSh=εh, ∀h,g ∈ H. “             (Majid 2002, Radford David E)
Proof
Let S and S1 be two antipodes for H. Then using properties of antipode, associativity of τ and co-associativity of Δ we get
S= τ∘(SâŠ-[ τ∘(IdâŠ-S1)∘Δ])∘Δ= τ∘(IdâŠ- Ï„)∘(S⨂IdâŠ-S1)∘(Id âŠ-Δ)∘Δ=
τ∘(τ⨂Id)∘(S⨂IdâŠ-S1)∘(Δ âŠ-Id)∘Δ = τ∘( [τ∘(S⨂Id)∘Δ]⨂S1)∘ Δ=S1.
So the antipode is unique.
Let Sâˆ-id=εs idâˆ-S=εt
To check that S is an algebra anti-homomorphism, we compute
S(1)= S(1(1))1(2)S(1(3))= S(1(1)) εt (1(2))= εs(1)=1,
S(hg)=S(h(1)g(1)) εt(h(2)g(2))= S(h(1)g(1))h(2) εt(g(2))S(h(3))=εs (h(1)g(1))S(g(2))S(h(2))=
S(g(1)) εs(h(1)) εt (g(2))S(h(2))=S(g)S(h), ∀h,g ∈H and we used εt(hg)= εt(h εt(g)) and εs(hg)= εt(εs(h)g).
Dualizing the above we can show that S is also a co-algebra anti-homomorphism:
ε(S(h))= ε(S(h(1) εt(h(2)))= ε(S(h(1)h(2))= ε(εt(h))= ε(h),
Δ(S(h))= Δ(S(h(1) εt(h(2)))= Δ(S(h(1) εt(h(2))⨂1)= Δ(S(h(1) ))(h(2)S(h(4))⨂ εt (h(3))=
Δ(εs(h(1))(S(h(3))⨂S(h(2)))=S(h(3))⨂ εs(h(1))S(h(2))=S(h(2))⨂ S(h(1)). (New directions)
Example2.2
“The Hopf Algebra H=Uq(b+) is generated by 1 and the elements X,g,g-1 with relations
gg-1=1=g-1g and g X=q X g, where q  is a fixed invertible element of the field k. Here
ΔX= X⨂1 +g ⨂ X, Δg=g ⨂ g, Δg-1=g-1⨂g-1,
εX=0, εg=1=ε g-1, SX=- g-1X, Sg= g-1, S g-1=g.
S2X=q-1X.”
Proof
We have Δ, ε on the generators and extended them multiplicatively to products of the generators.
ΔgX=(Δg)( ΔX)=( g⨂g)( X⨂1 +gâ