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Math 304 Solutions 5

Math 304 Solutions 5 - Math 304 Solutions 5 1 Section 3.6 6...

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Math 304 Solutions 5 1 Section 3.6 6. How many solutions will the linear system A x = b have if b is in the column space of A and the column vectors of A are linearly dependent. Explain. Answer: There will be infinitely many solutions. Proof. By Theorem 3.6.2 in the book (or by the theorem in class), if b is in the column space of A then the system A x = b is consistent. Thus, the system has either 1 solution or infinitely many solutions. Since the column vectors are linearly dependent, we know that if we row reduce the matrix A , there will be free variables. Thus, the system has either no solutions or infinitely many solutions. Therefore, the system has infinitely many solutions. 9. Let A and B be row equivalent matrices. (a) Show that the dimension of the column space of A equals the dimension of the column space of B . (b) Are the column spaces of the two matrices necessarily the same? Justify your answer. Answer: (a) Since A and B are row equivalent, they have the same row space. Thus, the dimension of the row space of A
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