Assignment 5 - MATH 223 Linear Algebra Winter 2010...

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MATH 223, Linear Algebra Winter, 2010 Assignment 5, due in class Monday, February 15, 2010 Reminder: the midterm exam is Thursday, March 4, from 6-7PM. The rooms in the Stewart Biology building are N2/2 and S1/3. 1. Let V = P 3 ( x ), the space of real polynomials with degree at most 3. Define T : V -→ V by T ( p ( x )) = ( x 2 - 2 x ) p 00 ( x ) + (2 x + 2) p 0 ( x ) - 12 p ( x ) . (a) Verify that T is a linear operator on V . (b) Find [ T ] B where B = (1 , x, x 2 , x 3 ) is the standard ordered basis for V . (c) Find [ T ] B 0 where B 0 = (1 , x +1 , x 3 - x 2 , x 2 ) is a nonstandard ordered basis for V . (d) Find a basis for each of ker ( T ) and im ( T ). (e) Show that T ( T + 12 I )( T + 10 I )( T + 6 I ) = 0. 2. Let V be the vector space M n ( F ) of all n × n matrices with entries from the field F ; let P V be any particular invertible matrix. Show that if we define TX = X - P - 1 XP for every X V , then T is a linear operator on V . For the rest of the problem, assume that n = 2, F = R and P = 1 1 1 2 . (a) Let B be the standard ordered basis for V , that is, B = 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 ¶¶ . Find [ T ] B . (b) Let B 0 be the following nonstandard basis for V . B 0 = 1 0 0 1 , 1 1 1 1 , 1 - 1 1 - 1 , 1 1 - 1 - 1 ¶¶ . Find [ T ] B 0 . (c) Find a basis for each of ker ( T ) and im ( T ). 3. Show the exchange property for linear span. That is, suppose that
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