ps7 - Problem Set 7 Math 115A/5 Fall 2011 Due Monday...

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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 0 ( 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). Problem 3 (5.2.20) Let W 1 , W 2 , . . . , W k be subspaces of a finite-dimensional vector space V such that k X i =1 W i = V . Prove that V is a direct sum of W 1 , W 2 , . . . , W k if and only if dim( V ) = k X i =1 dim( W i ) .
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