2114-Homework5.pdf

# 2114-Homework5.pdf

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Math 2114 Subspaces Homework 5 Name: 1. Let A = 3 - 1 0 1 - 2 - 9 3 - 3 - 6 15 0 3 - 1 1 2 - 5 3 23 2 2 8 - 18 - 1 2 , which has RREF B = 1 0 1 - 2 0 3 0 1 3 - 7 0 2 0 0 0 0 1 8 0 0 0 0 0 0 . Find a basis for the following subspaces: (a) Col A (b) Nul A (c) Row A (d) Col B (e) Nul B (f) Row B 2. Let P 4 be the set of polynomials of degree at most 4. Let H be the set: H = p ( x ) P 4 Z 1 0 p ( x ) dx = 0 and p (0) = p (1) and p 0 (0) = p 0 (1) (a) Let p ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 , translate the equation Z 1 0 p ( x ) = 0 into a linear equation in the a i ’s. (something like a 1 + 2 a 2 + 4 a 3 = 0) (b) Let p ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 , translate the equation p (0) = p (1) into a linear equation in the a i ’s. (something like a 1 + 2 a 2 + 4 a 3 = 0) (c) Let p ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 , translate the equation p 0 (0) = p 0 (1) into a linear equation in the a i ’s. (something like a 1 + 2 a 2 + 4 a 3 = 0) (d) Consider the three linear equations you found in (a)–(c), write those as a linear system: C~a = ~ 0.
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