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Unformatted text preview: FACT 3. If H is any subspace of a ﬁnitedimensional vector space V , then any set of linearly independent vectors in H can be extended to a basis for H , H is ﬁnitedimensional 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 = 12 47 0 9 1 2 5 1 0 0 0 13 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 , and12 t + 8 t 3 form a basis for P 3 . HOMEWORK: SECTION 4.5...
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 Spring '08
 PAVLOVIC
 Linear Algebra, Algebra, Equations, Vectors, Vector Space, Sets, basis, A. Example

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