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Unformatted text preview: Problem Set 7 Math 115A/5 — Fall 2011 Due: Monday, November 28 Please note a typo was corrected in problem 2. Problem 1 (5.2.3) For each of the following linear operators T on a vector space V , test T for diagonalizability, and if T is diagonalizable, find a basis β for V such that [ T ] β is a diagonal matrix. (a) V = P 3 ( R ) and T is defined by T ( f ( x )) = f ( x ) + f 00 ( x ). (b) V = P 2 ( R ) and T is defined by T ( ax 2 + bx + c ) = cx 2 + bx + a . (c) V = R 3 and T is defined by T a 1 a 2 a 3 = a 2 a 1 2 a 3 . (d) V = P 2 ( R ) and T is defined by T ( f ( x )) = f (0) + f (1)( x + x 2 ). (e) V = C 2 and T is defined by T ( z,w ) = ( z + iw,iz + w ). (f) V = M 2 × 2 ( R ) and T is defined by T ( A ) = A t . Problem 2 (5.2.7) For A = 1 4 2 3 ∈ M 2 × 2 ( R ) , find an expression for A n , where n is an arbitrary positive integer. (Hint: computing A n is easy when A is diagonal)....
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This note was uploaded on 04/09/2012 for the course MATH 172a taught by Professor Kong,l during the Fall '08 term at UCLA.
 Fall '08
 Kong,L
 Math, Vector Space

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