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HWSolutions1.4

# HWSolutions1.4 - Newberger Math 247 Spring 03 Homework...

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Newberger Math 247 Spring 03 Homework solutions: Section 1.4, #31-36 31. Let A be a 3 × 2 matrix. Explain why the equation A x = b cannot be consistent for all b in R 3 . Generalize your argument to the case of an arbitrary A with more rows than columns. Notice that saying the equation A x = b is consistent for all b in R 3 is the same thing as saying that the columns of A span all of R 3 . Theorem 4 says that the columns of A span all of R 3 only when A has a pivot position in every row. Since A is 3 × 2 there can be at most 2 pivots, so there cannot be a pivot in every row. Geometrically, we can see that the largest subset of R 3 that the 2 columns of A can span is a plane, so there will be some vectors b that will not lie in this plane, and the system A x = b will not always be consistent. Generally, if A has more rows than columns, there cannot be a pivot in every row, and the system A x = b will not always be consistent.

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HWSolutions1.4 - Newberger Math 247 Spring 03 Homework...

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