# Calculating the Values Based on Linear Algebra Questions

Description

Hi,I need stepwise solutions for all the questions in the attached document. Neatly write your answers and scan them in a word/pdf document.Thanks π

1. (16 pts.) Define addition on ππΓπ (β) by
π΄ β π΅ = β(π΄ + π΅)
and scalar multiplication by
π β π΄ = βππ΄
where π΄ and π΅ are in ππΓπ (β) and π is a real number and the operations of the right-hand side of
these equations are the usual ones associated with matrices. Determine which of the properties for a
vector space are satisfied on ππΓπ (β) with the operations β and β. Examine each property.
2. (16 pts.) For parts (a) and (b) consider the subset S of P2 given by
π = {2 + π₯ 2 , 4 β 2π₯ + 3π₯ 2 , 1 + π₯}.
(a) Determine whether the set S is linearly independent.
(b) Determine whether the set S spans P2 .
3
1
1
1
3. (8 pts.) Determine whether [β2] β Span { [0], [ 1 ], [ 4 ] }.
5
2
β1
β10
1 3
4. (26 pts.) Let π΄ = [3 10
2 5
β2
β4
β6
1
6 ]. Use the techniques discussed in video lecture to complete
β1
parts (a), (b), and (c). Show all steps!
(a) Find a basis for ππ(π΄).
(b) Find a basis for ππ(π΄).
(c) Find a basis for πΆπ(π΄).
5. (10 pts.) Determine whether the following sets S are subspaces of π2Γ2 (β).
(a) π = {π΄ β π2Γ2 (β)| π΄ is singular}
(b) π = {π΄ β π2Γ2 (β)| tr(π΄) = 0}
1
β1
β3
6. (8 pts.) Consider the following vectors from β3 : v1 = [β1], v2 = [ 2 ], v3 = [ 5 ]. Use the fact
1
2
6
that dim(β3 ) = 3 to determine whether v1, v2 , and v3 form a basis for β3 .
7. (16 pts.) Solve each differential equation and simplify your answer. Where indicated, find an explicit
solution.
(a)
(b)
ππ¦
ππ₯
ππ¦
ππ₯
+
+
2π₯π¦
π₯ 2 +2
= 0 (explicit solution)
2π₯π¦
π₯ 2 +2π¦
=0

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