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SAINT MARY’S UNIVERSITY

DEPARTMENT OF MATHEMATICS AND COMPUTING

SCIENCE

MATH2321.2 Linear Algebra II

FINAL EXAMINATION

April 11 – April 22, 2020

In Isolation Mode

Instructor: A. Finbow

Arthur Cayley

1821 – 1895

Sir William R. Hamilton

1805 – 1865

F. Georg Frobenius

1849 – 1917

Cayley in 1858 published Memoir on the theory of matrices in which, among other things, he proved that,

in the case of 2 × 2 matrices, a matrix satisfies its own characteristic equation. He stated that he had

checked the result for 3 × 3 matrices, indicating its proof, but said: “I have not thought it necessary to

undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree”. In

1853, Hamilton had already proven the 4 × 4 case in the course of his investigations into quaternions.

The general case was first proved by Frobenius (Schur’s Doctoral supervisor) in 1878.

Instructions

1. Questions require a detailed solution. Sketchy solutions or mere answers will

not suffice. Include all steps so that the examiner has a clear indication of

how you arrived at each solution.

2. There are 221 possible marks.

3. The work that you submit must be your own: collaboration and/or plagiarism

is not permitted.

5. Due via e-mail: April 22, 2020 by 11:59pm.

HAPPY EASTER!!

1.

[1 point] Create your personal 4-dimensional vector f by placing the last 4 digits of your

A-number consecutively in the 4 entries of f as shown in this example.

𝟓

EXAMPLE if you’re A-number is A12345678 then your personal vector is(𝟔).

𝟕

𝟖

2.

2

1

Let V be the subspace of R4 spanned by 𝐸 = {(1) , (0) , 𝐟}, where f is your personal 41

2

1

0

dimensional vector from question 1.

a) [12 points] Use the Gram-Schmidt method to find an orthonormal basis for V.

b)

3.

4.

[8 points]

2

Use your result in part a) to find the projection of the vector (3) on V.

2

1

4

7

1

Consider the set of vectors S = {(2) , (5) , (8)}. As you all know, this is a dependent set

3

6

9

in the vector space R3.

a)

3

[8 points] Decide if S is independent set in 𝐙11

(i.e. the 3-dimensional space in which

arithmetic is mode 11).

b)

3

[8 points] Decide if S is independent set in 𝐙13

(i.e. the 3-dimensional space in which

arithmetic is mode 13).

a)

[8 points] Decide if the set S = {1 + x + x2 + x3, 1 − x + 2×2 − x3, 1 − 2x + x2 + 4×3} is

independent in P3.

b)

[8 points] Decide if 1 + 2x + 3×2 + 4×3 is in span(S).

c)

[8 points] Suppose that P3 is equipped with the inner product:

2

= ∫−1 𝑓(𝑡)𝑔(𝑡)𝑑𝑡. Find the projection of x + 2 onto x2.

5.

[10 points] Find the Error

Trial Theorem. Let W be a subspace of Rn. Then the projection matrix from Rn onto W is

the identity matrix.

Proof:

Let P be the projection matrix from Rn onto W. By Theorem 7.11, P = A (ATA)1 T

A . But then, using Theorem 3.9 c) we have

P = A (ATA)-1AT = A(A-1(AT) -1)AT = (AA-1)((AT) -1AT) = (I)(I) = I

QED

6.

1

[10 points] Let W be the subspace of R4 spanned by 𝑆 = {(−1) , 𝐟}, where f is your

0

1

personal 4-dimensional vector from question 1. Find the 4 by 4 projection matrix P from Rn

onto W.

7.

a)

b)

8.

[10 points] Find the characteristic polynomial and the eigenvalue(s) for the matrix

1 −4

(

).

3 −2

8 −2 2

[15 points] Orthogonally diagonalize the following matrix B = (−2 5 4) [given

2

4 5

that its eigenvalues are and 9]. Your answer should consist of both an orthogonal

matrix Q and a diagonal matrix such that QTBQ = . Don’t multiply out this

product!

Let T: M22→ M22 be the linear transformation defined by

1 −2

T(𝑋) = 𝑋 (

).

−3 6

a) [8 points] Find a basis for Ker(T).

b) [8 points] Find a basis for Im(T) (same as Range(T)).

c) [2 points] Show that T2(X) = 7T(X).

1 2

)

−3 4

d) [2 points] Use the result in c) [even if you did not “get” part c)], to find T 2020 (

0

1

Let B = {(

0

2 0

0

),(

),(

0

0 0

0

3

0 0

),(

)} be the basis for M22.

0

0 4

1 2

) with respect to the basis B.

−3 4

e) [2 points] Find the coordinates of (

f)

[8 points] Find the matrix representing T with respect to the basis B.

𝑥 𝑦 𝑧

9. [9 points] Suppose that det( 𝑟 𝑠 𝑡 )= −2.

𝑎 𝑏 𝑐

In each part, evaluate the expression.

y

z

x

a) det 2r − x 2 s − y 2t − z =

a

b

c

𝑠 𝑏 𝑦

b) det( 𝑡 𝑐 𝑧 ) =

𝑟 𝑎 𝑥

c)

10.

𝑠

det(𝑠

𝑟

𝑏

𝑏

𝑎

𝑦

𝑦) =

𝑥

[10 points] Prove that if A and B are n by n matrices then det(AB) = det(BA).

11.

[2 points] Who was Schur’s Doctoral supervisor?

12.

[28 points] This problem relates directly with the proof of Schur’s Theorem. I am

assuming that you have the theorem in front of you and have read over the commentary

2

4

3

in the Notes for March 31 (part 1). I will take the matrix A = (−4 −6 −3) and tell

3

3

1

you that one of its eigenvalues is −2. You are to find a unitary (orthogonal in this case)

triangulation T for A following the method of the proof.

1. Compute a unit eigenvector u corresponding to −2.

2. Extend to an orthonormal basis for R3 with the first vector in the basis and form the

matrix U.

3. Multiply UTAU to find the 2 by 2 matrix B.

4. Find an eigenvalue for B and repeat the above steps to produce the matrix W

5. From this construct the matrix V, and obtain the product UV.

6. Finally write down the matrix T.

7. Now that you know all the eigenvalues, check if A is diagonalizable.

13. [10 points] Suppose that {v1, v2, v3} is an independent set in a vector space V. Prove that

{v1, v1 + 3v2, v1 + v2 + 2v3} is also independent.

SHORTER ANSWERS [26 Marks (2 for each part)]

In each part answer the question, evaluate the expression or indicate that there is not enough

information to proceed.

2−𝑖

3𝑖 − 1

(

)

(

a) Compute the complex inner product where 𝒖 =

and 𝒗 =

−𝑖 )

5𝑖

6𝑖 + 1

2

2−𝑖

b) Normalize ( −𝑖 ).

6𝑖 + 1

c)

3 − 2𝑖

𝑖

Compute (

∗

4

).

1 + 2𝑖

d)

2 4

)

4 1

If A is a matrix with entries in Z5, compute det(A).

e)

If A is a matrix with entries in Z7, compute det(A).

f)

Let T: Rn→P2020 (the vector space of 3 by 3 matrices) be a linear transformation. If T is one

to one, what are the possible values for n?

g)

Let T: Rn→ P2020 be a linear transformation. If T is onto, what are the possible values for n?

For the next 2 questions consider the matrix A = (

h) The Cayley-Hamilton Theorem states that a matrix satisfies its own characteristic equation.

Who was the person who proved the general case of this theorem and approximately how

old was he when the first proof of a special case was given?

i)

The 3 by 3 real symmetric matrix A has an eigenvalue = 2 of algebraic multiplicity one

and another eigenvalue of algebraic multiplicity two. A is also known to represent an

indefinite quadradic form and det(A) = 18. Find the value of .

For the next four questions suppose that A is a 3 by 3 matrix with eigenvalues 1 and 2.

Suppose also that the rank of A − I is equal to 1.

j) Which eigenvalue of A is repeated? EXPLAIN WHY

k) Write down a specific matrix that is similar to A and symmetric.

l) Write down a specific matrix that is similar to A and not symmetric. EXPLAIN WHY they

are similar.

m) Write down a specific matrix that has the same eigenvalues as A but is not similar to A.

EXPLAIN WHY they are not similar.

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