quiz2_S06 - Let A = 1 1 0 1 0 0 , ~ b = 1 1 1 . (a)...

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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.
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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 ).
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2. (15 points) Find the real eigenvalues of the matrix: A = 5 1 - 5 2 1 0 8 2 - 7 .
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3. (15 points) Find an orthonormal bases for the kernel of the matrix: A = ± 1 1 3 2 1 0 2 1 ² .
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4. (20 points)
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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 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 ....
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quiz2_S06 - Let A = 1 1 0 1 0 0 , ~ b = 1 1 1 . (a)...

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