Can you please help me with just A8W #4(b) and A9Q #6-10 True/False just answers please?  

 

Due: Friday, July 8 at 4:00PM 1 0
1 1 −2
4 1 1 .
1. Find a basis for the four fundamental subspaces of A = 3 1
1 −3 −2 2 2
2. Prove of disprove the following statement: There exists a matrix A ∈ M3×3 (R) such that Null(A) =
Col(A).
3. Let L : R3 → R3 be the linear mapping defined by
L(x1 , x2 , x3 ) = (2x1 − x2 , x2 + x3 , x3 − 3x1 ) 1
0 1
and consider the basis B = 1 , 0 , 1 for R3 . 0
1
0
(a) Find [L].
(b) Find [L]B
(c) Find a matrix P such that [L]B = P −1 [L] P
4. Let B and C both be bases for Rn and let L : Rn → Rn be a linear operator.
(a) Prove that [L]B and [L]C are similar.
(b) Prove that rank ([L]B ) = dim (Range(L)). 1

Math 636 Assignment 9 - Quiz Component 3 −4 2
1. Let A = 1 −2 2. Which of the following is not an eigenvector of A?
1 −5 5 1
1
2
1 (a) 1
(b) 1
(c) 1
(d) 2
2
1
1
1 1
2
1
2 ?
2. What is the geometric multiplicity of the eigenvalue λ = 2 of A = −2 6
3 −6 −1
(a) gλ = 1
(b) gλ = 2
(c) gλ = 3 3. Let
(a)
(b)
(c)
(d) 2 1 1
A = 0 1 0. Which of the following is not an eigenvalue of A?
2 0 1
λ=0
λ=1
λ=2
λ=3 3
1
2
2
2 . Which of the following is not an eigenvalue of B?
B= 2
−5 −1 −4
λ=1
λ = −1
λ=2
λ = −2 4. Let
(a)
(b)
(c)
(d) 1 2 5. Which of the following matrices is diagonalizable? 1
2
1
2
(a) −2 6
3 −6 −1 −3 2 1
(b) 0 −3 0
0
1 1 3 2
(c) 2 3 1 −1 −1 3 7
(d) 0 5 −2
0 0 −8 For questions 6 - 10, determine if the statement is True or False. You should make sure that
you have a proof of each true statement and a counter example for each false statement.
6. If A is diagonalizable, then A and its reduced row echelon form R have the same
eigenvalues.
(a) True.
(b) False.
7. The columns of an n × n matrix A are linearly dependent if and only if λ = 0 is an
eigenvalue of A.
(a) True.
(b) False.
8. If A and B are diagonalizable, then A + B is diagonalizable.
(a) True.
(b) False.
9. If A is invertible, then A is diagonalizable.
(a) True.
(b) False.
10. If A is diagonalizable, then A is invertible.
(a) True.
(b) False.

Attachments:

• 1492840207-A8W.pdf
• 1492840207-A9Q.pdf