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20f_95s_fe

# 20f_95s_fe - Math 20F Bender Final Exam 3 PM Remember to...

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Math 20F Bender Final Exam 3 PM June 15, 1995 Please start each problem on a NEW PAGE. Remember to show your work. 1. (25 pts.) Define the following: (a) an eigenvalue of the n × n matrix A , (b) the null space of an m × n matrix A , (c) the dimension of a vector space V . 2. (15 pts.) Show that T ( x 1 , x 2 , x 3 ) = (3 x 2 - x 3 , x 1 - 4 x 2 , x 3 ) is a linear transformation and find its standard matrix. 3. (25 pts.) Find the eigenvalues and eigenspaces of 1 1 2 0 2 - 1 0 0 - 1 . 4. (20 pts.) Let W be the span of the orthogonal vectors 1 1 0 - 1 , 1 0 1 1 and 0 - 1 1 - 1 . Write 3 4 5 6 as the sum of a vector in the subspace W and a vector orthogonal to W . 5. (20 pts.) Find the eigenvalues associated with the quadratic form 8 x 2 1 + 6 x 1 x 2 and use them to classify the form. 6. (60 pts.) Suppose that A is an n × n matrix and that R is the reduced row echelon form of A . You are given R but are not given A . For each of the following explain why you can or cannot answer it given R but not A . (a) Does A - 1 exist? (b) What is a basis for Nul A ? (c) What is a basis for Col
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