Turn in problems 1 & 2.

1. Give an example of a 3 x 3 matrix A such that the first  two rows of rref(A) are nonzero rows and the third row is all zeros, but row(A) is not spanned by the first two rows of A.
2. Give an example of a 3 x 4 matrix A such that the leading columns of rref(A) do not span col(A).
3. Let A be any matrix. Prove that the columns of rref(A) are linearly dependent if and only if the columns of A are linearly dependent. You may not use the Fundamental Theorem of Invertible Matrices, nor the fact that row operations do not change linear dependence relations between columns --- the point of the problem is to prove this very fact. Hint: The columns of A are linearly dependent iff Ax=0 has a nontrivial solution.

These problems are for review purposes. They aren't  directly related to Section 5.1. For the rest of the semester, I will every now and then assign a few "review" problems in order to refresh your memory and solidify your understanding of some of the older topics.