Test3 - 2 T ) 3. Find the matrix T M of 4 2 : R R T ,...

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TEST 3 (Math 250 A) 1. (a) Define what it means for a linear map W V T : to be an isomorphism. (30 pts) (b) Suppose that dim V = dim W . Then show that a linear map W V T : is an isomorphism if and only if it is one-to-one or onto. ( Hint : Use the Dimension Formula) (c) Show that the following linear map is an isomorphism: ) , , , ( ) ( : 4 2 2 d c c b c b a d c b a T R R M T ( Hint : Use part (b) above) 2. (a) True or False : If the linear map W V T : is an isomorphism, then T (30 pts) sends a basis of V to a basis of W . (b) Let 2 2 : R R T be linear, defined by: ) , ( ) , ( x y x y x T . Explain why T has an inverse 1 T . Find a formula for 1 T . ( Hint : It will be useful to see where T sends the natural basis of 2 R )
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(c) (i) Does the linear map 4 4 : R R T , defined by: ) , , , ( ) , , , ( x t t z z y y x t z y x T have an inverse? ( Hint : Look at the Ker( T ) ) (ii) For the linear map 2 2 : R R T , defined by ) , ( ) , ( y x ky y x T , R k , show that T T 1 . ( Hint : It would be easier if you computed
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Unformatted text preview: 2 T ) 3. Find the matrix T M of 4 2 : R R T , defined by ) , 4 , 2 3 , 2 ( ) , ( x y y x y x y x T , relative to the bases B = {(1,1), (1,0)} and D = natural basis of 4 R . Use the matrix to compute T ( u ) , where u = (1,2). (10 pts) ( Hint : Make sure that u is expressed with respect to the appropriate basis before you pass it through the matrix T M . The u given above is with respect to the natural basis of 2 R ) 4. Let 2 2 : R R T be linear, defined by ) , ( ) , ( y x y x T , and consider the (30 pts) bases B = natural basis of 2 R and D = {(1,1), (-1,0)}. Find: (a) The matrix T M of T relative to the basis B. (b) The transition matrix P from basis B to basis D. (c) Use P above to find the matrix T N of T relative to the basis D....
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This test prep was uploaded on 04/18/2008 for the course MAT 250 taught by Professor Aristidou during the Spring '07 term at DigiPen Institute of Technology.

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Test3 - 2 T ) 3. Find the matrix T M of 4 2 : R R T ,...

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