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SECTION 4.5 DIMENSION 1) Must every basis for R 23 have 23 vectors? 2) What are the possible sets of solutions of a homogeneous linear system of equations in four variables? In other words, what kind of subspaces does R 4 have? FACT 1. If a vector space V has a basis with exactly n vectors, then any set of more than n vectors in V must be linearly dependent. WHY? Use the coordinate mapping to move everything into R n , where we already know that a set of more than n vectors must be linearly dependent. FACT 2. If a vector space has a basis with exactly n vectors, then every basis of this vector space has exactly n vectors. DEFINITION. If a vector space V is spanned by a finite set of vectors, then V is finite- dimensional, and the number of vectors in a basis for V is called its dimension, written dim V . A vector space that cannot be the span of any finite set of vectors is called infinite- dimensional.
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EXAMPLES. (1) R p (2) a + b 3 b 5 a - 2 b - 4 a
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Unformatted text preview: FACT 3. If H is any subspace of a finite-dimensional vector space V , then any set of linearly independent vectors in H can be extended to a basis for H , H is finite-dimensional and dim H ≤ dim V . FACT 4. If we already know that dim V = p , then (1) any set of p linearly independent vectors is a basis for V , and (2) any set of p vectors that span V is a basis for V . EXAMPLE. Find the dimensions of Nul A and Col A when A = 1-2 4-7 0 9 1 2 5 1 0 0 0 1-3 2 0 0 0 1 0 0 0 0 . FACT 5. The dimension of Nul A is the number of free variables in the equation A x = , and the dimension of Col A is the number of pivot columns in A . EXAMPLE. Show that the first four Hermite polynomials 1, 2 t ,-2 + 4 t 2 , and-12 t + 8 t 3 form a basis for P 3 . HOMEWORK: SECTION 4.5...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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