I learned invertible matrix theorem, and I want to know the relation of this two statements.
(k) There is an $m\times n$ matrix $D$ such that $AD=I$
(g) The equation $Ax=b$ has at least one solution for each b in $\Bbb{R}^n$
Book says (k) implies (g) but there is no proof so I want to prove this. Statement (g) is the definition of "onto", however I cannot prove this. How can I derive (g) from (k)?
For this you simply need the associativity of matrix multiplication. Then, if you take $x=Db$ you have $A(Db)=(AD)b=Ib=b$
The intuition here is that if $Ax=b$ then "$x=A^{-1}b$" - you don't necessarily have an inverse, but $D$ has the property of the inverse that you do need, and will do in place of $A^{-1}$.