This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 onetoone 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 ....
View
Full
Document
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.
 Spring '09
 RUDYAK

Click to edit the document details