finsol2000

finsol2000 - M346 Final Exam Soutions Problem 1 Consider...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M346 Final Exam Soutions 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 . We compute: L 1 b 1 = 1 2 3 5 = 2 = 2 b 2 L 1 b 2 = 1 2 3 5 = 3 5 = 3 b 1 + 5 b 2 L 1 b 3 = 1 2 3 5 = 2 = 2 b 4 L 1 b 4 = 1 2 3 5 = 3 5 = 3 b 3 + 5 b 4 so we have [ L 1 ] B = 3 2 5 3 2 5 b) Find the matrix of L 2 relative to the same basis. We compute: L 2 b 1 = 2 3 5 1 = 3 = 3 b 3 L 1 b 2 = 2 3 5 1 = 3 = 3 b 4 L 1 b 3 = 2 3 5 1 = 2 5 = 2 b 1 + 5 b 3 L 1 b 4 = 2 3 5 1 = 2 5 = 2 b 2 + 5 b 4 so we have [ L 2 ] B = 2 2 3 5 3 5 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? The eigenvalues are λ 1 = 3 / 4 and λ 2 =- 7 / 4, with eigenvectors b 1 = (3 , 7) T and b 2 = (1 ,- 1) T . The first mode is stable since | λ 1 | < 1, while the second is unstable since | λ 2 | > 1. The dominant eigenvector is λ 2 =- 7 / 4. 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? Now the question is whether the real part of λ is positive or negative....
View Full Document

This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas.

Page1 / 4

finsol2000 - M346 Final Exam Soutions Problem 1 Consider...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online