m235notes16 - Notes 16: Vector Spaces: Bases, Dimension,...

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Unformatted text preview: Notes 16: Vector Spaces: Bases, Dimension, Isomorphism Lecture November, 2011 Let V be a vector space. Definition 1. Let v 1 ,v 2 , v m V . A linear combination of the elements v i is any element of V of the form m 1 a i v i ,a i R . Definition 2. Let S V . The span of S is the set of all linear combinations of elements of S. If W V is a subspace of V , we say that S spans W if the span of S is W. Definition 3. Let S V . We say that S is linearly independent if, whenever m 1 a i v i = ,a i R ,v i V all of the a i = 0 . Definition 4. Let W be a subspace of V and S a subset of W . We say S is a basis of W provided S spans W and S is linearly independent. Theorem 1 . Let S be a basis of a subspace W of V . We can write every element w W uniquely as w = a i v i ,a i R ,v i S. We call the coefficients a i the coordinates of w with respect to S. Proof. Since S spans W , we can write w = a i v i ,a i R ,v i S. Assume that we an write w in two different ways: w = X a i v i = X b i v i ,a i ,b i R ,v i S. We obtain 0 = w- w = X ( a i- b i ) v i . Since S is linearly independent, a i = b i . Thus the expressions for w are the same. Definition 5. Let W be a subspace of V . The dimension of W is the number of elements in a basis of W . If a basis of W is infinite we say that the dimension is infinite....
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m235notes16 - Notes 16: Vector Spaces: Bases, Dimension,...

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