1.6 Basis and Dimensions 1

1.6 Basis and Dimensions 1 - 1.6. 1.6. Bases and Dimension...

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Unformatted text preview: 1.6. 1.6. Bases and Dimension 1. (a) F ( ) is a basis for the zero vector space. * span { } = { } and is linearly independent. (b) T ( ) Theorem 1.9 ; If a vector space V is generated by a finite set S , then some subset of S is a basis for V . (c) F (Counterexample) { 1 ,x,x 2 , } is a basis for P ( F ) (d) F ( ) Corollary 2 (c) from Theorem 1.10 Every linearly independent subset of V can be extended to a basis for V . (e) T ( ) Corollary 1 from Theorem 1.10 Let V be a vector space having a finite basis. Then every basis for V contains the same number of vectors. (f) F ( ) { 1 ,x,x 2 , ,x n } is a basis for P n ( F ) (g) F ( ) The dimension of M m n ( F ) is m n (h) T ( ) Replacement theorem. (i) F ( ) Theorem 1.8. Let V be a vector space and = { u 1 ,u 2 , ,u n } be a subset of V . Then is a basis for V if and only if each v V can be uniquely expressed as a linear combination of vectors of . (Example) V = R 2 , S = { v 1 = (1 , 0) ,v 2 = (0 , 1) ,v 3 = (1 , 1) } ( a,b ) = av 1 + bv 2 + 0 v 3 = 0 v 1 + ( b- a ) v 2 + av 3 32 PNU-MATH 1.6. (j) T Theorem 1.11. Let W be a subspace of a finite-dimensional vector space V . Then W is finite-dimensional and dim(W) dim(V). Moreover, if dim(W)=dim(V), then V = W ....
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1.6 Basis and Dimensions 1 - 1.6. 1.6. Bases and Dimension...

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