FinalTestReviewSpring2011

But not augmented form that is we need to ignore the

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Unformatted text preview: Answer: The most general solution of A￿ = w with c = 12 is x2 = −2 + x3 2 . x￿ x3 0 1 (c) The basis is formed by pivot rows of either A or its reduced form. (But not augmented form, that is, we need to ignore the last column in our calculation.) Thus, the basis can be given either as {(1, 1, 0), (0, 2, 4)} or as {(1, 0, −2), (0, 1, 2)}. 1 Ex. 3. Let A be an 7 × 11 matrix with the rank r = 4. What is the dimension of the following four spaces. (a) Column space C (A) of A. (b) Row space C (AT ) of A. (c) Null space N (A) of A. (d) Left-null space N (AT ) of A. ( pts) Solution: Answers: (a) = r = 4. (b) = r = 4. (c) = n − r = 11 − 4 = 7. (d) = m − r = 7 − 4 = 3. Name “left-null” comes from the fact that y is in N (AT ) when it is a solution of AT ￿ = 0, which is equivalent to (AT ￿ )T = 0T , that is, to ￿ T A = 0. y y y Ex. 4. Let A be an m × n matrix of rank r. Describe precisely the possible number of solutions of a system A￿ = ￿ under the following assumptions: xb (a) r < m and...
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