Homework 12 Solution

Description

1. A linear transformation T : R³ ! R³ 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

0. • 9

without finding the matrix for T .

0. Identify the image of following vectors under the transformation T . Be sure to justify your conclu-sions.

2 3 2 3

• 1=2

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

• 9=2

(b) Explain why T	02	35	31	=	3	2	35	3	.
	@4	37	5A			4	37	5

1. 1. 2. 31

1

1. 3. Calculate T @4 25A. 12

2. Let T : R² ! R² be the linear transformation that reflects the entire R² across the x-axis.

2. 1. Without calculating a matrix A for the transformation T , determine what the eigenvectors and eigen-values would be, if any. In other words, does the transformation have any stretch directions and associated stretch factors? Justify your answer.

2. 2. Find a matrix A to represent the transformation T . Calculate its eigenvectors and associated eigen-values for the matrix A, and verify your answers to part (a).

• 3

1 1 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 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.

1. 2. Compute A¹⁰⁰ by diagonalizing A.

4. A is an n n matrix.

4. 1. (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		.

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

2. 1. as a hint to prove the following statements.

2. 1. 1. ² is an eigenvalue of A². (Also review problem #6 of Homework 11.)

      2. If A is invertible, ¹ is an eigenvalue of A ¹.

      3. + 1 is an eigenvalue of A + I.