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AMS210-02Spring09 PracticeTest1

AMS210-02Spring09 PracticeTest1 - AMS 210.03 David Fried...

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AMS 210.03 David Fried Test 1 B Name_________________________________ No calculators are permitted. Complicated arithmetic expressions may be left as is and need not be evaluated. 1. The matrix A = 2 5 6 1 has eigenvectors u 1 =[1,1] and u 2 =[-5,6] a) What are the corresponding eigenvalues λ 1 and λ 2 ? (10 points) b) Write the vector v =[7,-15] as a linear combination of u 1 and u 2 . (10 points) c) Use your result from (b) to determine A 10 v . (5 points) 2. Let the matrices A and B and the vectors x and y be given by: A = 1 0 3 1 0 2 2 0 B = 3 0 1 2 x =[1,-1] y =[3,0,2] a) Which of the following products are possible (you need not compute them and you may use the vectors as either row vectors or column vectors)? (5 points) x∙y, AB, BA, xA, Ax, yB, Ay b) Perform one of the possible multiplications from part (a). (5 points) 3. Consider the following growth model for the interaction between caterpillars and wasps in a garden: C′ W′ = 1.3 .5 . 5 . 8 C W What vector norm measures the total bug population? Use the corresponding matrix norm to bound the possible growth in the total population over a period of 3 months. (10 points)
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