com07sA - AMS Common Exam - Part A, January 2007 Name: ID...

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Unformatted text preview: AMS Common Exam - Part A, January 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! Section 1: Linear Algebra Choose three of the four problems to solve. 1. Consider the 2 2 matrix: A = a b b- a (a) Find a general expression for A m , m N ( m is any natural number). (b) Extend this result to A q , q Z ( q is any integer). For what values of a and b is this result valid? 2. Consider the linear transformation, F : R 3 R 3 defined by: F ( x, y, z ) = (2 x- 3 y- 2 z,- 3 x + 3 y- 3 z,- 2 x-...
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com07sA - AMS Common Exam - Part A, January 2007 Name: ID...

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