Calculating the Values Based on Linear Algebra Questions

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 🙂

1. (16 pts.) Define addition on 𝑀𝑚×𝑛 (ℝ) by
𝐴 ⊕ 𝐵 = −(𝐴 + 𝐵)
and scalar multiplication by
𝑐 ⊗ 𝐴 = −𝑐𝐴
where 𝐴 and 𝐵 are in 𝑀𝑚×𝑛 (ℝ) and 𝑐 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 𝑀𝑚×𝑛 (ℝ) with the operations ⊕ and ⊗. Examine each property.
2. (16 pts.) For parts (a) and (b) consider the subset S of P2 given by
𝑆 = {2 + 𝑥 2 , 4 − 2𝑥 + 3𝑥 2 , 1 + 𝑥}.
(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 [−2] ∈ Span { [0], [ 1 ], [ 4 ] }.
5
2
−1
−10
1 3
4. (26 pts.) Let 𝐴 = [3 10
2 5
−2
−4
−6
1
6 ]. Use the techniques discussed in video lecture to complete
−1
parts (a), (b), and (c). Show all steps!
(a) Find a basis for 𝑁𝑆(𝐴).
(b) Find a basis for 𝑅𝑆(𝐴).
(c) Find a basis for 𝐶𝑆(𝐴).
5. (10 pts.) Determine whether the following sets S are subspaces of 𝑀2×2 (ℝ).
(a) 𝑆 = {𝐴 ∈ 𝑀2×2 (ℝ)| 𝐴 is singular}
(b) 𝑆 = {𝐴 ∈ 𝑀2×2 (ℝ)| tr(𝐴) = 0}
1
−1
−3
6. (8 pts.) Consider the following vectors from ℝ3 : v1 = [−1], v2 = [ 2 ], v3 = [ 5 ]. Use the fact
1
2
6
that dim(ℝ3 ) = 3 to determine whether v1, v2 , and v3 form a basis for ℝ3 .
7. (16 pts.) Solve each differential equation and simplify your answer. Where indicated, find an explicit
solution.
(a)
(b)
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
+
+
2𝑥𝑦
𝑥 2 +2
= 0 (explicit solution)
2𝑥𝑦
𝑥 2 +2𝑦
=0