Description
1. A linear transformation T : R^{3} ! R^{3} has an eigenvector 4 0 5 associated with eigenvalue 1=4 and two
2
2 3 2 3

1
eigenvectors 4 15 and 4 15 both associated with eigenvalue 3. Answer all of the following questions


9

without finding the matrix for T .

Identify the image of following vectors under the transformation T . Be sure to justify your conclusions.
2 3 2 3

1=2
(i) 4 0 5 (ii) 4 1=25

9=2

(b) Explain why T
02
3_{5}
31
=
3
2
^{3}5
3
.
@4
37
5A
4
37
5



31


1


Calculate T @4 25A. 12


Let T : R^{2} ! R^{2} be the linear transformation that reflects the entire R^{2} across the xaxis.


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 associated stretch factors? Justify your answer.



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
1 1 1
3. Let A = 4_{1} 
1 
1 
5. 
1 
1 
1 


(Strang x5.2 #3) Without solving det(A I) = 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 eigentheory?
Hint 2. Note that the rows of A add up to the same number 3, which would lead you to another eigenvectoreigenvalue pair.


Compute A^{100} by diagonalizing A.


A is an n n matrix.


(Strang x5.1 #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 is an eigenvalue of A, the way to find an associated eigenvector is to
.

(Strang x5.1 #24) Let be an eigenvalue of A with associated eigenvecter ~x. That is, A~x = ~x. Use part


as a hint to prove the following statements.




^{2} is an eigenvalue of A^{2}. (Also review problem #6 of Homework 11.)

If A is invertible, ^{1} is an eigenvalue of A ^{1}.

+ 1 is an eigenvalue of A + I.

