Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH 2090 Problem Workshop Week 5

1. Let S = {v1, v2, . . . , vr} and suppose W = span(S). Show that if x is orthogonal to all vectors in S, then x is orthogonal to all vectors in W.

2. If W is the subspace of by W = {(0, a, 0, b, 0) | a, b ∈ R} ,determine and its dimension.

3. If possible. Diagonalize the following matrices:

4. Suppose A and B are similar matrices, show that

(a) If A is invertible, then so is B.

(b) They have the same nullity. Hint: Show if Q is invertible, then the null spaces of QB and B are the same.

(c) They have the same rank

(d) They have the same trace

5. Suppose k is a positive integer. Show that if λ is an eigenvalue of A with eigenvector x, then is an eigenvalue of with eigenvector x.

6.      (a) Show that the characteristic equation of a 2 × 2 matrix A can be expressed as 

(b) Suppose that the characteristic equation of a 2×2 matrix is  Show that

(c) Use part (b) to compute