*This preview shows
pages
1–2. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **M346 Final Exam December 13, 2000 Problem 1. Consider the vector space M 2 , 2 of 2 2 matrices, let B = 2 3 5 . Con- sider the linear transformations L 1 ( A ) = AB and L 2 ( A ) = BA . a) Find the matrix of L 1 relative to the basis 1 , 1 , 1 , 1 . b) Find the matrix of L 2 relative to the same basis. Problem 2. Let A = 1 4- 4 3 7 . a) Consider the equations x ( n ) = A x ( n- 1), with A as above. What are the stable and unstable modes? What is the dominant eigenvector? b) Consider the equations x ( t ) = A x ( t ), with A as above. What are the stable and unstable modes? What is the dominant eigenvector? Problem 3. Let A = 1 5 3 4 4- 3 . Which of the following are Hermitian? Which are unitary? Which are both? Which are neither? a) A b) A + I c) e A d) e iA Problem 4. In R 4 with the standard inner product, consider the vectors b 1 = (1 , , , 1) T , b 2 = (1 , 2 , 2 , 1) T , b 3 = (2 , 1 , 1 , 0) T , b 4 = (1 , 3 , 5 , 7) T ....

View
Full
Document