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Exam1A200

# Exam1A200 - AD = I p Explain why the equation A x = b has a...

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M340L EXAM 1A 2:00 FALL, 2009 Dr. Schurle Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . . . . 1. (16 points) After many row operations you have changed the augmented matrix of a system of linear equations to the following, which is not yet in reduced echelon form. Describe all solutions of the system in parametric vector form. DO NOT USE A CALCULATOR!! SHOW ALL YOUR WORK STEP BY STEP!!! 1 0 4 2 0 2 2 4 28 16 1 11 1 3 19 11 0 5 0 - 2 - 10 - 6 1 1

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YOUR SCORE: /100 2. (14 points) Give three different properties of a list of vectors, each equivalent to the list of vectors being linearly dependent. Then choose two of your properties and explain why they are equivalent, that is, explain why a list of vectors that has each one of the properties must also have the other. 3. (14 points) Suppose A and D are p × p matrices such that

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Unformatted text preview: AD = I p . Explain why the equation A x = b has a solution for every b in R p and then explain what this says about the columns of A . 4. (14 points) Do the columns of the following matrix span R 4 ? Justify your answer. 1 2 1 2 1 2 8 7 16 4 1 5 4 11 4-2-1-5-2 5. (14 points) Find the inverse of A = " 4 3 3 2 # and use it to solve AX = " 2-2 3 1 0 5 # 6. (14 points) Find the general ﬂow pattern of the network shown in the ﬁgure. Assuming that the ﬂows are all nonnegative, what is the largest possible value for x 3 ? 7. (14 points) Suppose the linear transformation T from R 2 to R 3 maps u = " 5 3 # into 2 3-1 and v = " 8 5 # into 7 2 1 (a) Find T (3 u + 5 v ) (b) Show that Span { u , v } = R 2 . (c) Use parametric vector form to describe the range of T ....
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