I was just watching Strang's test review for Unit 1 and he made a comment about the null space in question 4. I want to check if it's generally true.
Given a matrix $A_{m\times n}$ over a field and it's reduce row echelon form, $R_{m\times n}$ such that we have
$\begin{bmatrix} I_{r\times r} & F_{r\times n-r} \\ 0 & 0 \end{bmatrix}$
Where $r$ is rank(A), then do the non-zero rows of
$\begin{bmatrix} I_{r\times r} & -F_{r\times n-r} \\ 0 & 0 \end{bmatrix}$
always form a basis for the null space?