Due: 11:59pm ET, Monday March 22, 2021

MAT224 Linear Algebra II

Assignment 4

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1. Let

1(a) Show that W is a subspace of .

1. Let

1(b) Determine dim W.

2. Let V be a vector space, and let T .

2(a) Prove that if T is injective, then 0 is not an eigenvalue of T.

2(b) Prove that if 0 is not an eigenvalue of T then T is injective.

3. Let V be a finite dimensional vector space, and let S, T ∈ L(V ).

3(a) Suppose that if λ 0 is an eigenvalue of ST. Show that λ is also an eigenvalue of T S.

3. Let V be a finite dimensional vector space, and let S, T .

3(b) Suppose that 0 is an eigenvalue of ST. Show that 0 is also an eigenvalue of T S.