Give an example of a 3 x 3 matrix A such that rref(A) has exactly two
nonzero rows (with the third row all zeros), but row(A) is not spanned by
the first two rows of A.
Give an example of a 3 x 4 matrix A such that the leading columns of
rref(A) do not span col(A).
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. Hint: The columns of
A are linearly dependent iff Ax=0 has a nontrivial solution.
Suppose A is a 3 x 4 matrix. Prove that the 2nd column of A is a linear
combination of its 1st and 4th columns iff the same is true for rref(A).
Hint: erase the 3rd column, then proceed as in the last problem.