final-sp94 - MAS 4105 Final Spring 1994 Show all your work...

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Unformatted text preview: MAS 4105 Final Spring 1994 Show all your work on the paper provided. Your work should be written in a proper and coherent manner. There are 7 compulsory problems giving a total of 80 points. 1. [10 pts] (a) Define subspace . (b) Let C 2 [0 , 1] denote the vector space of functions that have a continuous second derivative on the interval [0 , 1]. Prove or disprove: The set V = { f C 2 [0 , 1] : f 00 ( x ) + f ( x ) = 0 } is a subspace of C 2 [0 , 1]. (c) Prove or disprove: The set W = { ( x,y ) R 2 : xy = 0 } is a subspace of R 2 . 2. [10 pts] (a) Define linear transformation . (b) Prove or disprove: L : C [0 , 1]- R defined by L ( f ) = s Z 1 [ f ( x )] 2 dx is a linear transformation. (c) Prove or disprove: T : P 2- P 3 defined by T ( p ( x )) = (1 + x ) p ( x ) + x 2 p ( x ) is a linear transformation. 3. [10 pts] (a) Define linear independence . (b) Suppose x 1 , x 2 ,..., x k are linearly independent vectors in R n and A is an invertible n n matrix. Prove that the vectorsmatrix....
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final-sp94 - MAS 4105 Final Spring 1994 Show all your work...

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