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Unformatted text preview: M 340L (Fall 2007, 60150) Review Problem Set Posted: Thursday, November 29, 2007 (Do NOT submit.) 1 FINAL EXAMINATION Date: Friday, December 14 Time: 2 5 PM Location: BEL 328 Disclaimer The final examination will cover all lectures, and Problem Sets 1 through 10, including recommended exercises. These review exercises are intended to assist you with your study, but they in no way limit the scope or types of questions that may appear in the final examination. 1. Find all solutions to the equation A x = , where A = 1 4 2 3 5 1 1 1 4 . 2. Suppose the augmented coefficient matrix of a cer tain system of linear equations is rowequivalent to the following matrix. 1 1 2 5 1 3 4 6 1 a) How many equations are there in the original system? How many unknowns? b) Express in parametric form the set of solu tions. ( Choose your own variable names; do not forget to indicate which variable corresponds to which column. ) 3. Consider the vectors in R 2 illustrated in the follow ing diagram: w v 1 v 2 @ @ @ I v 3 a) Can w be written as a linear combination of v 1 and v 2 ? If so, can it be done uniquely? Briefly explain. b) Can w be written as a linear combination of v 2 and v 3 ? If so, can it be done uniquely? Briefly explain. c) Can w be written as a linear combination of v 1 , v 2 , and v 3 ? If so, can it be done uniquely? Briefly explain. 4. Determine whether each of the following sets of vec tors are linearly independent: a) v 1 = @ 1 5 1 A , v 2 = @ 1 2 8 1 A , and v 3 = @ 4 1 1 A b) v 1 = @ 1 7 6 1 A , v 2 = @ 2 9 1 A , v 3 = @ 3 1 5 1 A , and v 4 = @ 4 1 8 1 A . c) v 2 = @ 2 3 5 1 A , v 3 = @ 1 A , and v 4 = @ 1 1 8 1 A . d) v 1 = B B @ 2 4 6 10 1 C C A , and v 2 = B B @ 3 6 9 15 1 C C A . M 340L (Fall 2007, 60150) Review Problem Set Posted: Thursday, November 29, 2007 (Do NOT submit.) 2 5. Let T : R 4 R 3 be the linear map defined by T ( x ) := 1 4 8 1 2 1 3 5 x . Determine if T is onetoone, and if T is onto. 6. Let T : R 2 R 2 be the linear map which first re flects points through the vertical axis and then ro tates points counterclockwise by / 2 radians. Find the standard matrix of T ....
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 Fall '08
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