Proof subject: the rank $=$ # of rows $\rightarrow$ the system of linear equation(SLE) is consistent
Description:
- I want to prove this deductively but not inductively.
- I have read the proof of a SLE is consistent $\iff$ rank$([\textbf {A}])=$ rank$([\textbf{A}|~\textbf{b}])$. It uses a concept of dimension. Like, It's in the dimension of $\textbf A$, so the system must be consistent because $\textbf b$ is image of $T_\textbf A$. But I want other proofs without this dimension thing. or I want someone could give a proof unfolding this relation between dimension and consistency because for what I've concerned, it's not straightforward enough as a proof by just saying if it's in the dimension of $\textbf A$, then blablabla....
- I have read a proof like: all of situation where the SLE is inconsistent must have a last zero row after eliminated into reduced Echelon Form. It's a more inductive way and definitely not what I want.