Four Fundamental Spaces Question Why do the four fundamental sub spaces come from Ax=b and A'y=f? Why these two equations? 

The first key observation is that if $x=(x_1,x_2,...,x_n)'$ and $y=(y_1,y_2,...,y_n)$ are two column vectors then 
$$x'y=x_1y_1+x_2y_2+...+x_ny_n=:x\cdot y$$
which is the standard scalar product. 
The second key observation is that if $A$ is an $m\times n$ matrix, and $z=(z_1,z_2,...,z_m)'$  is a column vector of dimension $m$, then
$$(Ax)\cdot y=x\cdot (A'y)$$
The four spaces get related in this equation. If $x\in\text{Ker}(A)$ then $Ax=0$. Therefore from the equation above $$x\cdot(A'y)=(Ax)\cdot y=0\cdot y=0$$
from where we see that $x$ is orthogonal to $A'y$ for all $y$. This shows that $\text{Ker}(A)$ is orthogonal to $\text{Im}(A')$. Other relations follow similarly.
