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math 235 hw5 - Homework 5 Due April 1 Adam Gamzon Related...

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Homework 5, Due April 1 Adam Gamzon Related reading: sections 3.4, 4.1, and 4.2 from the text 1. Let γ = 1 2 , 5 6 and let β = 2 5 , 5 12 . These are bases of R 2 . (You don’t need to prove this.) (a) Let v = - 4 4 . Compute [ v ] γ . Solution: Solve the system represented by the matrix: 1 5 | - 4 2 6 | 4 . This has rref 1 0 | 11 0 1 | - 3 , so [ v ] γ = 11 - 3 . (b) Determine the change of basis matrix, [1] βγ , that takes γ -coordinates to β -coordinates. Solution: The change of basis matrix is 2 5 5 12 - 1 1 5 2 6 = - 12 5 5 - 2 1 5 2 6 = - 2 - 30 1 13 . (c) Use part (b) to find [ v ] β . Solution: [ v ] β = [1] β γ [ v ] γ = - 2 - 30 1 13 11 - 3 = 68 - 28 2. Consider the vector space P 2 = { ax 2 + bx + c : a, b, c R } . (Remember, the symbol means “in.”) (a) Show that β = { 1 + x 2 , - 3 x, 2 x + x 2 } is a basis of P 2 . 1
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Solution: Since dim P 2 = 3, it is enough to show that these three vectors are linearly independent. That is, we’ll show that the only real numbers a, b, and c such that a (1 + x 2 ) + b ( - 3 x ) + c (2 x + x 2 ) = 0 are a = b = c = 0. Looking at the equation, we get a system of linear equations a = 0 ( - 3 b + 2 c ) x = 0 ( a + c ) x 2 = 0 , or equivalently, a = 0 - 3 b + 2 c = 0 a + c = 0 .
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