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

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Unformatted text preview: Homework 5, Due April 1 Adam Gamzon Related reading: sections 3.4, 4.1, and 4.2 from the text 1. Let = 1 2 , 5 6 and let = 2 5 , 5 12 . These are bases of R 2 . (You dont need to prove this.) (a) Let v =- 4 4 . Compute [ v ] . 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 ] = 11- 3 . (b) Determine the change of basis matrix, [1] , that takes -coordinates to -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 ] . Solution: [ v ] = [1] [ v ] =- 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 = { 1 + x 2 ,- 3 x, 2 x + x 2 } is a basis of P 2 . 1 Solution: Since dim P 2 = 3, it is enough to show that these three vectors are linearly independent. That is, well show that the only real numbersindependent....
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This note was uploaded on 11/22/2011 for the course MATH 255 taught by Professor Staff during the Spring '10 term at UMass (Amherst).

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

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