2.1 Linear Transformations 2

2.1 Linear Transformations 2 - 2.1. 20. (a) Let w1 , w2 T...

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20. (a) Let w 1 ,w 2 T ( v 1 ) , a F then v 1 ,v 2 V 1 s.t. w i = T ( v i ) , i = 1 , 2 So aw 1 + w 2 = aT ( v 1 ) + T ( v 2 ) = T ( av 1 + v 2 ) T ( v 1 ) T ( v 1 ) is a subspace of W (b) Let K = { x V | T ( x ) W 1 } ,a F Let x 1 ,x 2 ı K s.t. T ( x 1 ) = w 1 ,T ( x 2 ) = w 2 W 1 T ( ax 1 + x 2 ) = aT ( x 1 ) + T ( x 2 ) = aw 1 + w 2 W 1 ax 1 + x 2 K, a F { x V | T ( x ) W 1 } is a subspace of V 21. (a) (i) T ( c ( a 1 ,a 2 , ··· )+( b 1 ,b 2 )) = T ( ca 1 + b 1 ,ca 2 + b 2 , ··· ) = ( ca 2 + b 2 ,ca 3 + b 3 , ··· ) = c ( a 2 ,a 3 ··· ) + ( b 2 ,b 3 , ··· ) = cT ( a 1 ,a 2 , ··· ) + T ( b 1 ,b 2 , ··· ) (ii) U ( c ( a 1 ,a 2 , ··· ) + ( b 1 ,b 2 )) = U ( ca 1 + b 1 ,ca 2 + b 2 , ··· ) = (0 ,ca 1 + b 1 ,ca 2 + b 2 , ··· ) = (0 ,ca 1 ,ca 2 , ··· ) + (0 ,b 1 ,b 2 , ··· ) = cU ( a 1 ,a 2 , ··· ) + U ( b 1 ,b 2 , ··· ) (b) (i) T is not one-to-one 0 6 = (1 , 0 , 0 , ··· ) N ( T ) (ii) T is onto R ( T ) = { ( a
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This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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2.1 Linear Transformations 2 - 2.1. 20. (a) Let w1 , w2 T...

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