Fundamental Theorem of Linear Algebra

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For any $$A \in \mathbb{R}^{m\times n}$$, it holds that
 * $$\text{rank}(A)+\dim\mathcal N (A) = n$$

This can be interpreted as conversation of dimension. That is, $$\text{rank}(A)$$ is the dimension of set 'hit' by the mapping $$y=Ax$$; $$\dim\mathcal N(A)$$ is the dimension of the set of $$x$$ 'crushed' to zero by $$y=Ax$$. Those two dimensions must always add up to $$n$$.

TBC