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com07fA - AMS Common Exam Part A May 2007 Name Part A Part...

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AMS Common Exam - Part A, May 2007 Name: ID Num. Part A: / 75 Part B: / 75 Total: / 150 This component of the exam (Part A) consists of two sections (Linear Algebra and Advanced Calculus) with four problems in each. Each question is worth 25 points; choose THREE questions to answer from EACH section. Each problem should be solvable in approximately 20 minutes or less. Provide your answer in the space provided, and show all work. If extra sheets are used, place them inside the booklet and note on the cover page how many additional pages are included. Good Luck!
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Section 1: Linear Algebra Choose three of the four problems to solve. 1. Given the linear transformation, F : R 4 R 3 defined by: F ( w, x, y, z ) = ( w + 3 x + 6 y + z, w + x - y - 2 z, 3 w + 5 x + 4 y - 3 z ) (a) Find a matrix representation of F . (b) Find the dimension of, and a basis for, both the kernel and the image of F .
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A = 0 1 0 1 0 1 0 1 0 (a) Find the eigenvalues of A . (b) Find matrices,
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