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

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2.1 Independence and Dimension 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.1. Independence and Dimension Defnition 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 1 + r 2 2 + ··· + r k k = ~ 0 with at least one r j 6 = 0. If such a dependence relation exists, then { 1 2 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 { 1 2 k } of W is a basis for W if and only if (1) W =sp( 1 2 k )and (2) the vector 1 2 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 deFnitions of basis and linear independent . Theorem. Finding a Basis for W = sp ( ~w 1 ,~w 2 ,..., ~w k ) . ±orm the matrix A whose j th column vector is j . If we row-reduce A to row-echelon form H , then the set of all 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 Sets. Let W be a subspace of R n .L e t 1 2 k be vectors in W that span W and let
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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

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