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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 β, γ, and δ 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 ] γ β is invertible. T F ii. Let T : V → W be a linear transformation and x ∈ V ; then [ T ( x )] γ = [ T ] γ β [ x ] β . 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 ] δ β = [ S ] δ β [ T ] γ β . 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 ( γ )). 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 β = { 1 , x, x 2 } be an ordered basis for P 2 ( R ) and let γ = 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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 Spring '09
 RUDYAK
 Linear Algebra, Vector Space, WI, linear transformation

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