Math 54 - Quiz 04 Ans, Spring 2006 (2)

Math 54 - Quiz 04 Ans, Spring 2006 (2) - 1 x x 2 x 3 x 4 x...

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Math 54, Quiz 4 Solutions Sections 202/203 GSI Carter 1. Let V be the real vector space of all polynomials of degree 7 or less. Let W be the set of all polynomials p in V such that p (4) = 0, which is a subspace of V . For each of the following statements, write the word “true” or “false.” (a) Any basis for W can be extended to a basis for V . True. This is true about subspaces in general, by a theorem in 3.6. (b) Any basis for V has some subset which is a basis for W . False. For a counterexample, consider the basis
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Unformatted text preview: { 1 , x, x 2 , x 3 , x 4 , x 5 , x 6 , x 7 } of V , none of whose elements are in W . 2. Find a basis for the null space of the following matrix. ± 2 3 2 2 4 3 3 4 ² The matrix row reduces as follows: ± 2 3 2 2-3-1 ² ± 2 1 2-3-1 ² ± 2 1 2 3 1 ² Therefore the null space consists of vector of the form (-a 2-b,-a 3 , a, b ) , so a basis of the null space is given by ³(-1 2 ,-1 3 , 1 , ) , (-1 , , , 1) ´ . 1...
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This note was uploaded on 02/20/2010 for the course AS a taught by Professor 11 during the Spring '10 term at École Normale Supérieure.

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