quiz2_S06

# quiz2_S06 - Let A = 1 1 0 1 0 0 ~ b = 1 1 1(a Determine ~x...

This preview shows pages 1–7. Sign up to view the full content.

Name Class Time MATH 202 - QUIZ # 2 Due April 21, 2006 at 2PM Time: 60 minutes Please show all work. Unsupported answers will receive no credit. Books, notes, calculators, are not permitted on this quiz. As part of your obligations under the Honor Code, do not discuss this quiz with anyone until after Friday, April 21 at 2PM. WRITE OUT AND SIGN THE PLEDGE: I pledge my honor that I have not violated the Honor Code during this examination.

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

View Full Document
1. (15 points) You are given a 2 × 2 matrix A with two linearly independent eigenvectors ~v 1 ,~v 2 with corresponding eigenvalues λ 1 > 0 and λ 2 = 0. Consider the dynamical system ~x ( t + 1) = A ~x ( t ) with an initial condition ~x (0) = 2 ~v 1 - 3 ~v 2 . Describe and sketch the trajectory of this dynamical system (for positive values of t ). (Hint: You must distinguish cases based on the value of λ 1 ).
2. (15 points) Find the real eigenvalues of the matrix: A = 5 1 - 5 2 1 0 8 2 - 7 .

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

View Full Document
3. (15 points) Find an orthonormal bases for the kernel of the matrix: A = ± 1 1 3 2 1 0 2 1 ² .
4. (20 points)

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: Let A = 1 1 0 1 0 0 , ~ b = 1 1 1 . (a) Determine ~x * which minimizes k A~x-~ b k among all ~x ∈ R 2 . (b) Let S be an orthogonal 3 × 3 matrix. Determine the least-square solution of the linear system SA ~x = S ~ b . 5. (a) (15 points) Let ~v 1 , ~v 2 be a bases of R 2 and T : R 2 → R 2 a linear transformation with T ( ~v 1 ) = ~v 1 + ~v 2 T ( ~v 2 ) = ~v 1-~v 2 . Determine the matrix of the 4-fold composition T 4 in the standard bases of R 2 . (b) (5 points) Consider the 2 n × 2 n block matrix A = ± 1 n-1 n ² where 1 n stands for the n × n identity matrix. Is A invertible? 6. (15 points) You are given a non-invertible 3 × 3 matrix A with three eigenvalues. You are told that the diagonal elements of A are a 11 =-2 , a 22 = 0 , a 33 = 7. Prove that at least one of the eigenvalues of A must be greater or equal to 5 2 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

quiz2_S06 - Let A = 1 1 0 1 0 0 ~ b = 1 1 1(a Determine ~x...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online