# prac2 - MAS 4105—Practice Problems for Exam#2 1 Define T...

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Unformatted text preview: MAS 4105—Practice Problems for Exam #2 1. Define T : P 2 ( R ) → P 2 ( R ) by T ( f ) = ( x + 1) f ′ ( x )- f ( x ). (a) Prove that T is a linear transformation. (b) Find a basis for R ( T ). (c) Find a basis for N ( T ). 2. (a) Let β = { 1 ,x } be a basis for P 1 ( R ) and let γ = { (1 , 2) , (2 , 0) } be a basis for R 2 . Suppose T : P 1 ( R ) → R 2 is a linear transformation such that [ T ] γ β = bracketleftbigg 1 2 3 4 bracketrightbigg . Compute T (3- x ). (b) Let V be a vector space with basis β = { vectorx 1 ,...,vectorx n } . Define T : V → F n by T ( vectorv ) = [ vectorv ] β . Prove that T is a linear transformation. 3. Let V and W be vector spaces with dim( V ) = n and dim( W ) = m . Let T : V → W be a linear transformation. (a) Define the rank of T and the nullity of T . (b) Give a formula which relates rank( T ) to nullity( T ). (c) Prove that if T is invertible then V and W have the same dimension....
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prac2 - MAS 4105—Practice Problems for Exam#2 1 Define T...

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