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Math416_ColumnNull

# Math416_ColumnNull - Math 416 Abstract Linear Algebra Fall...

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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 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 dim Col 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 Span { a 1 , a 2 , a 3 } = Span { a 1 , a 3 } and a 2 can be discarded.

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