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 ππΓπ (β) by

π΄ β π΅ = β(π΄ + π΅)

and scalar multiplication by

π β π΄ = βππ΄

where π΄ and π΅ are in ππΓπ (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 ππΓπ (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 [β2] β Span { [0], [ 1 ], [ 4 ] }.

5

2

β1

β10

1 3

4. (26 pts.) Let π΄ = [3 10

2 5

β2

β4

β6

1

6 ]. Use the techniques discussed in video lecture to complete

β1

parts (a), (b), and (c). Show all steps!

(a) Find a basis for ππ(π΄).

(b) Find a basis for π
π(π΄).

(c) Find a basis for πΆπ(π΄).

5. (10 pts.) Determine whether the following sets S are subspaces of π2Γ2 (β).

(a) π = {π΄ β π2Γ2 (β)| π΄ is singular}

(b) π = {π΄ β π2Γ2 (β)| tr(π΄) = 0}

1

β1

β3

6. (8 pts.) Consider the following vectors from β3 : v1 = [β1], v2 = [ 2 ], v3 = [ 5 ]. Use the fact

1

2

6

that dim(β3 ) = 3 to determine whether v1, v2 , and v3 form a basis for β3 .

7. (16 pts.) Solve each differential equation and simplify your answer. Where indicated, find an explicit

solution.

(a)

(b)

ππ¦

ππ₯

ππ¦

ππ₯

+

+

2π₯π¦

π₯ 2 +2

= 0 (explicit solution)

2π₯π¦

π₯ 2 +2π¦

=0