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Unformatted text preview: Math 416  Abstract Linear Algebra Fall 2011, section E1 Column space and null space The following example illustrates the notion of dimension and “culling down” a linearly depen dent collection of vectors. Let A = a 1 a 2 a 3 = 1 2 1 2 4 1 3 6 1 1 2 1 . Find the dimension of Col A and Null A , as well as a basis for each. A = 1 2 1 2 4 1 3 6 1 1 2 1 ∼ 1 2 1 0 0 1 0 0 2 0 0 ∼ 1 2 1 0 0 1 0 0 0 0 0 0 ∼ 1 2 0 0 0 1 0 0 0 0 0 0 Since the pivots are in columns 1 and 3, we conclude dimCol A = 2 and a basis of Col A is given by { a 1 ,a 3 } = { 1 2 3 1 , 1 1 1 1 } . Remark 1: We have culled down the linearly dependent collection { a 1 ,a 2 ,a 3 } to a basis of Span { a 1 ,a 2 ,a 3 } = Col A . In other words, since a 2 is already in Span { a 1 ,a 3 } , we have...
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This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.
 Fall '11
 FRANKLAND
 Math, Linear Algebra, Algebra, Vectors

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