2009_Winter_Practice_Midterm_1

2009_Winter_Practice_Midterm_1 - x 1 v 1 x 2 v 2 x 3 v 3 =...

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Problem 1. True or False: For each statement below, determine whether it is true or false, and circle the appropriate letter. You do not need to justify your answer. (5 points each) ( T F ) If A and C are matrices (of the appropriate sizes) and AC = 0 then A = 0 or C = 0. ( T F ) If A = [ a 1 a 2 a 3 a 4 ] is a 3 × 4 matrix and the set { a 1 , a 2 , a 4 } is linearly independent, then the equation A x = b has a solution for every b in R 3 ( T F ) If A and B are ( n × n ) matrices such that A has a column of all zeros, then AB has a column of all zeros. ( T F ) If A is an m × n matrix and there exist vectors x 1 and x 2 in R n such that A x 1 = A x 2 , then for any b such that the equation A x = b has a solution, the solution is not unique.
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Problem 2. Let v 1 = 3 0 0 - 1 , v 2 = 0 1 2 - 1 , v 3 = 3 0 2 0 . a) (20 points) Find a solution to the equation
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Unformatted text preview: x 1 v 1 + x 2 v 2 + x 3 v 3 = 3-1-6-2 . b) (10 points) Is the solution you found above unique? Why or why not? Problem 3. (15 points each) Let A = 1-2-3 1 4 2-3 4 . a) Find A-1 . b) Use A-1 to find the solution to A x = 2-2 Problem 4. (10 points each) a) Give an example of a 4 × 3 matrix A such that the solution set to A x = is a line. b) Give an example of a 4 × 3 matrix A such that A x = b has a unique solution for every b of the form b = b 1 b 3 b 4 and say why your example works....
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2009_Winter_Practice_Midterm_1 - x 1 v 1 x 2 v 2 x 3 v 3 =...

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