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c2s1 - 1 2.1 Independence and Dimension Chapter 2 Dimension...

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2.1 Independence and Dimension 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.1. Independence and Dimension Definition 2.1. Let { w 1 , w 2 , . . . , w k } be a set of vectors in R n . A dependence relation in this set is an equation of the form r 1 w 1 + r 2 w 2 + · · · + r k w k = 0 with at least one r j = 0. If such a dependence relation exists, then { w 1 , w 2 , . . . , w k } is a linearly dependent set. A set of vectors which is not linearly dependent is linearly independent . Theorem 2.1. Alternative Characterization of Basis Let W be a subspace of R n . A subset { w 1 , w 2 , . . . , w k } of W is a basis for W if and only if (1) W = sp( w 1 , w 2 , . . . , w k ) and (2) the vector w 1 , w 2 , . . . , w k are linearly independent.
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2.1 Independence and Dimension 2 Note. The proof of Theorem 2.1 follows directly from the definitions of basis and linear independent . Theorem. Finding a Basis for W = sp ( w 1 , w 2 , . . . , w k ) . Form the matrix A whose j th column vector is w j . If we row-reduce A to row-echelon form H , then the set of all w j such that the j th column of H contains a pivot, is a basis for W . Example. Page 134 number 8 or 10. Example. Page 138 number 22. Theorem 2.2. Relative Sizes of Spanning and Independent
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