Solving Ax = 0 , pivot variables, free varaibles, need more specific explanation

Question

Recently i started to lear linear algebra out of MIT OpenCourceWare. But i cant quite understand one little 'proof' (demontration) of getting $N(A)$ out of expression $Ax = 0$ by creating rref form of matrix. Given point is at minute 28:05 in this video

So basically what i cant understand is that..

We have $Ax = 0$ in form of :

$$ \begin{bmatrix} I & F \\ 0 & 0 \end{bmatrix}\begin{bmatrix} -F \\ I \end{bmatrix}=0 $$

Also the matrices sizes are :

$ r = rank(A) \\ A^{n\times m} \\ I^{r\times r} \\ F^{r\times (n-r)} $

So i really cant grasp this idea when there is $ (n-r) \neq r$ because then we cant express second matrix $x$ (because those blocks of I and F are not of the same width...)

So I want to ask, where to find better explanation of this ??

Answer 1

Those two $I$'s don't need to be the same size, the one in the $2 \times 2$ block matrix is $r \times r$ and the one in the block column matrix is $(n - r) \times (n - r)$.