midterm2solution

midterm2solution - Name TA Math 20F Linear Algebra Midterm...

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Name: PID: TA: Sec. No.: Sec. Time: Math 20F: Linear Algebra. Midterm Exam 2 February 16, 2011 Turn off and put away your cell phone. You may use one page of notes, but no books or other assistance during this exam. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. If you run out of space on a page, you should continue writing on the back of the same page. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. Question Points Score 1 6 2 6 3 6 4 6 5 6 Total: 30
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1. (6 points) Suppose A is a 4 × 3 matrix with rank 3. (a) Does the problem Ax = b have a solution for every b R 4 ? Explain your answer completely. Solution: No. Since dim Col A = 3 < 4 = dim R 4 , the columns of A cannot span all of R 4 . Alternatively, since the reduced echelon form of A has only 3 pivots, it must have one row of zeros, and hence certain choices of b may give inconsistent equations (like 0 = 1). (b) If Ax = b has a solution for some b R 4 , is that solution unique? Explain your answer completely. Solution: Yes. Since A has rank 3, all of its columns are pivots (i.e., they are linearly independent). Hence, the nullity is 0, and any solution is unique.
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