a3 - 1 2 3 3 = 2 1 2 3 3 Show that A is unique(b If A is...

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McGill University Math 236: Algebra 2 Assignment 3: due Wednesday, February 15, 2006 Midterm Test: Friday, February 17 (in class). 1. (a) If D is the differentiation operator on V = C R ( R ) and a R , prove that Ker((D - a) n ) = Span(e ax , xe ax ,..., x n - 1 e ax ) . (b) If W is the solution space of the differential equation f iv ( x ) - 2 f 00 ( x ) + f ( x ) = 0, show that W = Ker((D - 1) 2 ) Ker((D + 1) 2 ). (c) Find the solution of the differential equation in (b) satisfying f (0) = 1 , f 0 (0) = 2 , f 00 (0) = 3 , f 000 (0) = 4 . 2. Let T be the linear operator on V = R 2 × 2 defined by T ( A ) = 2 A + A t , where A t is the transpose of A . (a) Find bases for the kernel and image of T . (b) Show that T 2 - 4 T + 3 = 0 and use this to find the eigenvalues of T . (c) Find a basis of V consisting of eigenvectors of T . 3. (a) Find a 4 × 4 real matrix A such that A 1 1 1 1 = 1 1 1 1 , A 1 2 1 2 = 1 2 1 2 , A 1 1 2 2 = 2 1 1 2 2 , A
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Unformatted text preview: 1 2 3 3 = 2 1 2 3 3 . Show that A is unique. (b) If A is the matrix in (a), find the eigenvalues of A and a basis for each eigenspace of A . 4. (a) Find a 4 × 4 real matrix A whose null space and column space are spanned by the column matrices 1 2 3 4 , 4 3 2 1 . (b) What are the eigenvalues of the matrix A found in (a)? Does R 4 × 1 have a basis consisting of eigenvectors of A ? 5. Let A = 1 2 2 2 2 1 2 2-2-1-2-2-2-1 . (a) Using the fact that A 2 = I , find the eigenvalues of A and a basis for each eigenspace of A . (b) Find an invertible matrix P such that P-1 AP is a diagonal matrix....
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