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Unformatted text preview: MAS 3114Practice Problem Set #1 1. (a) Use Gaussian elimination to convert the matrix below to reduced echelon form. Each step in your computation should use only one elementary row operation. bracketleftbigg 1 2 4 1 2 1 2 8 1 2 bracketrightbigg (b) Use (a) to determine the solution to the system x 1 + 2 x 2 4 x 3 + x 4 = 2 x 1 2 x 2 + 8 x 3 x 4 = 2 . Your solution should express the pivot variables in terms of the free variables. (c) Let vectorx 1 ,vectorx 2 , . . . ,vectorx k be vectors in R n . Give the definition of Span { vectorx 1 ,vectorx 2 , . . . ,vectorx k } . 2. (a) Let A = bracketleftbigg 2 1 6 3 bracketrightbigg and vector b = bracketleftbigg b 1 b 2 bracketrightbigg . Determine the set of all vector b such that the system Avectorx = vector b has a solution. (b) Define what it means for the set S = { vectorx 1 ,vectorx 2 , . . . ,vectorx k } R n to be linearly independent. (c) Give an example of a set S = { vectorx 1 ,vectorx 2 ,vectorx 3 } R 3 such that Span( S ) negationslash = R 3 and no vector in S is a scalar multiple of any other vector in S . 3. (a) Give an example of a linear transformation T : R 2 R 3 such that T ( vectore 1 ) negationslash = vector and T is not onetoone. (b) Let T : R 2 R 2 be a linear transformation such that T parenleftbiggbracketleftbigg 1 1 bracketrightbiggparenrightbigg = bracketleftbigg 2 3 bracketrightbigg and T parenleftbiggbracketleftbigg 1 bracketrightbiggparenrightbigg = bracketleftbigg...
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This note was uploaded on 12/27/2011 for the course MAS 3114 taught by Professor Olson during the Fall '08 term at University of Florida.
 Fall '08
 Olson

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