## Question

###### 3. (15 pts) Let A be an m x n matrix with rank r, and let...

3. (15 pts) Let A be an m x n matrix with rank r, and let V = C(A). (a) V CIRP for what p? (b) What is V. in terms of a fundamental subspace for A? (c) How many vectors are in a basis for V, and how many in a basis for v 1? (d) For what m, n, and r docs Ax=b have a solution for every b? (e) Is a set of r vectors in V necessarily lincarly dependent? How about +1 vectors?

## Answers

A is a mxn matrix with rank r, let, v=CCA) (a) vcIRP Then, v=1RM P=m (6) vt=NCAT) = left null space of A. er since, rank A=r, There are rbarts vector in the basis of v, and mor in the basis of vt. d) If m=n=r Ax=b have a solution for every.b ... (e) no r vectors in v are not necessary dependent. but rtl rectors in v are necessary dependent, since, dim v =n.

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