CommonAJan09 - AMS Common Exam - Part A, January 2009 Name:...

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AMS Common Exam - Part A, January 2009 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. Consider the linear system of real variables: kx + y + 2 z = 0 kz + ky + z = p k 2 z + k 2 y + 2 kz = q (a) For what values of k, p, q R will the system have (i) one, (ii) many, of (iii) no solution? (b) When there is a unique solution, what is it? (c) When there are many solutions, what is the general solution in parametric form? What is the dimension and a basis for the associated homogenous system?
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2. Consider the linear map
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CommonAJan09 - AMS Common Exam - Part A, January 2009 Name:...

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