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Exam1B - x in R 3 for which T x = 5(16 points After many...

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M340L EXAM 1B SPRING, 2010 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. (14 points) Describe in detail the method used to determine whether p vectors in R q are linearly independent and explain why the method works.
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YOUR SCORE: /100 2. (14 points) Suppose A and D are matrices, not necessarily square, such that 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 . 3. (14 points) Find the inverse of D = " 2 1 3 2 # and use it to solve X - " 1 2 3 4 #! D = " 2 2 1 3 # for X .
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4. (14 points) Suppose the linear transformation T from R 3 to R 2 maps x = x 1 x 2 x 3 into x 1 " 3 9 # + x 2 " 1 4 # + x 3 " 2 2 # . (a) Calculate and simplify T 1 2 3 . (b) Find a matrix A such that T ( x ) = A x . (c) Use parametric vector form to describe the collection of all
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Unformatted text preview: x in R 3 for which T ( x ) = . 5. (16 points) After many row operations you have changed the matrix A to the following, which is not yet in reduced echelon form. Describe all solutions of the system A x = in parametric vector form. DO NOT USE A CALCULATOR!! SHOW ALL YOUR WORK STEP BY STEP!!! 3 2 13 1 24 11-2 1-4-3-3 1 0 3 4 2 1 0 3-1 2-1 6. (14 points) Do the columns of the following matrix span all of R 3 ? Justify your answer. 1 3 8 4-2-2-17-6 4 35 10 7. (14 points) Write the coefficient matrix that results from using the vector equation ap-proach to balance the following chemical reaction. DO NOT SOLVE THE SYSTEM!! NaHCO 3 + H 3 C 6 H 5 O 7 → Na 3 C 6 H 5 O 7 + H 2 O + CO 2...
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