laex2 - MAS 4105 Test 2 1. (8 pts) In the following let V,...

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Unformatted text preview: MAS 4105 Test 2 1. (8 pts) In the following let V, W, and U be a vector spaces of finite dimension over a field F with ordered bases , , and respectively. Indicate whether the following are true or false. i. If T : V W is linear and dim( V ) = dim( W ), then T is invertible if and only if [ T ] is invertible. T F ii. Let T : V W be a linear transformation and x V ; then [ T ( x )] = [ T ] [ x ] . T F iii. If T : V W is linear, then T is one-to-one if and only if T is onto. T F iv. If T : V W and S : W U are linear, then [ ST ] = [ S ] [ T ] . T F v. If T : V W and S : W U are linear and if x Nul( T ), then x Nul( ST ). T F vi. Let T : V W be a transformation, then T is linear if and only if T ( ax + y ) = aT ( x )+ T ( y ) for all x, y V and a F . T F vii. Let T : W U be a linear transformation, then R ( T ) = span( T ( )). T F viii. Let S : U W be an invertible linear transformation, then nullity( S- 1 ) + rank( S- 1 ) = dim( U ). T F 2. (10 pts) Let T : R 3 R 3 be defined by T a b c = a + b + c a + b c . i. Prove that T is linear. ii. What is the Null space for T and what is its dimension? iii. Use the Dimension Theorem to determine the rank of T . 3. (15 pts) Let = { 1 , x, x 2 } be an ordered basis for P 2 ( R ) and let = 1 , 1 1 , 1 1 1 be an ordered basis for R 3 . Define the invertible, linear transformation T : P 2 ( R ) R 3 by T ( ax 2 + bx + c ) = a + c a + b a ....
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This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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laex2 - MAS 4105 Test 2 1. (8 pts) In the following let V,...

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