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tut3 - MATH 1104F Solutions to Tutorial Assignment 3...

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MATH 1104F - Solutions to Tutorial Assignment 3 February 2, 2010 1. Let T : R 2 R 2 be the linear transformation that rotates points through - 3 π/ 4 (clockwise). (a) Find the standard matrix for T . (b) Find T ( x 1 , x 2 ). Solution: (a) T ( e 1 ) = ( - 2 / 2 , - 2 / 2) and T ( e 2 ) = ( 2 / 2 , - 2 / 2). Thus the standard matrix for T is A = [ T ( e 1 ) T ( e 2 ) ] = [ - 2 / 2 2 / 2 - 2 / 2 - 2 / 2 ] . (b) A [ x 1 x 2 ] = [ - 2 / 2 2 / 2 - 2 / 2 - 2 / 2 ] [ x 1 x 2 ] = [ - 2 2 x 1 + 2 2 x 2 - 2 2 x 1 - 2 2 x 2 ] Hence T ( x 1 , x 2 ) = ( - 2 2 x 1 + 2 2 x 2 , - 2 2 x 1 - 2 2 x 2 ). 2. Let T : R 3 R 4 be the linear transformation defined by T ( x, y, z ) = ( x + 2 z, y - 3 z, x + 2 y - 4 z, - x + y - 5 z ) . (a) Find the image of (1 , 1 , 1). (b) Find a vector v in R 3 whose image under T is ( - 1 , 5 , 9 , 6). (c) Is there more than one v whose image under T is ( - 1 , 5 , 9 , 6)? (d) Determine if (1 , 0 , 1 , 1) is in the range of the transformation T . (e) Find the standard matrix for T . (f) Is T one-to-one? Explain. (g) Is T onto? Explain. Solution: (a) T (1 , 1 , 1) = (3 , - 2 , - 1 , - 5) (b) 1 0 2 | - 1 0 1 - 3 | 5 1 2 - 4 | 9 - 1 1 - 5 | 6 R 3 R 3 - R 1 R 4
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