In Exam 1 the very last question - in the space of all 3x3 matrices we should consider the subspace of all rref's...I suppose one of the possible answers is that rref's do not form a subspace, since they are not closed under multiplication (that would break the 'leading-ones-rule')...

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anonymous · Answer

The point is that the RREF matrices themselves do not themselves form a subspace, but that they *span* a subspace.

You're absolutely right that, say (and I use $2\times2$ matrices for brevity)
$$\mathrm{B}=\left\{\begin{bmatrix}1&0\0&1\end{bmatrix},\begin{bmatrix}1&1\0&0\end{bmatrix},\begin{bmatrix}1&0\0&0\end{bmatrix}\right\}$$is not a subspace because this set of matrices is not closed under multiplication. But the question isn't concerned with whether $\mathrm{B}$ itself is a subspace.

Instead, we're interested with what subspace of all $2\times2$ matrices is spanned by $\mathrm{B}$ (and hence what a basis for this subspace might be - it happens to be $\mathrm{B}$).