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math222final - Math 222 Fall 2004 Final Exam Closed books...

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Math 222, Fall 2004: Final Exam Closed books, no notes, no calculators. Upper case letters denote matrices, boldface lower case letter vectors, and plain lower case letters scalars. The transpose of a matrix A is denoted A t . The identity matrix is denoted I . A matrix U is said to be orthogonal if U t U = I . In questions 2 – 6, please provide brief explanations for the steps in your calculations. Question 1: There is no connection between the matrices and the vectors in the following questions. No motivation is required. (2p each) (a) Suppose that A is an m × n matrix of rank k . What is dim(Col( A ))? (b) Suppose that there exists a non-zero vector y such that A y = 0. Does there exist a vector b such that the equation A x = b has a unique solution? (c) Is it true that λ is an eigenvalue of A if and only if det( A - λI ) = 0? (d) Is it true that all square matrices are diagonalizable? (e) What conditions must a matrix satisfy for it to have a singular value decomposition?
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