# MATH 141: Linear Analysis I Homework 12 solved

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1. A linear transformation T : R
3 → R
3 has an eigenvector

−1
0
2

 associated with eigenvalue 1/4 and two
eigenvectors

0
−1
5

 and

1
−1
9

 both associated with eigenvalue −3. Answer all of the following questions
without finding the matrix for T.
(a) Identify the image of following vectors under the transformation T. Be sure to justify your conclusions.
(i)

3
0
−6

 (ii)

1/2
−1/2
9/2

(b) Explain why T

3
−5
37

 = −3

3
−5
37

.
(c) Calculate T

−1
−2
12

.
2. Let T : R
2 → R
2 be the linear transformation that reflects the entire R
2 across the x-axis.
(a) Without calculating a matrix A for the transformation T, determine what the eigenvectors and eigenvalues would be, if any. In other words, does the transformation have any stretch directions and
(b) Find a matrix A to represent the transformation T. Calculate its eigenvectors and associated eigenvalues for the matrix A, and verify your answers to part (a).
3. Let A =

1 1 1
1 1 1
1 1 1

.
(a) (Strang §5.2 #3) Without solving det(A−λI) = 0, use observation to find all eigenvalues of A and then
find associated eigenvectors.
Hint 1. What can you say about the rank of A and what does that tell you about the nullspace? What
does nullspace have to do with eigen-theory?
Hint 2. Note that the rows of A add up to the same number 3, which would lead you to another
eigenvector-eigenvalue pair.
(b) Compute A100 by diagonalizing A.
4. A is an n×n matrix.
(a) (Strang §5.1 #23) Fill in the blanks.
i. If you know ~x is an eigenvector of A, the way to find the associated eigenvalue is to .
ii. f you know λ is an eigenvalue of A, the way to find an associated eigenvector is to .
(b) (Strang §5.1 #24) Let λ be an eigenvalue of A with associated eigenvecter ~x. That is, A~x = λ~x. Use part
(a) as a hint to prove the following statements.
i. λ
2
is an eigenvalue of A2
. (Also review problem #6 of Homework 11.)
ii. If A is invertible, λ
−1
is an eigenvalue of A−1
.
iii. λ + 1 is an eigenvalue of A + I.