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# prac1 - MAS 3114—Practice Problem Set#1 1(a Use Gaussian...

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Unformatted text preview: MAS 3114—Practice 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 one-to-one. (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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prac1 - MAS 3114—Practice Problem Set#1 1(a Use Gaussian...

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