I am struggling like 6 hours to understand what this content in the middle mean. Can u get me some clue to interpret it?
I understand all the stuff on the sides. So, row space is perp. to null space and they complement each other in R^m (by complement I mean that linear combination of bases both spaces span R^m) and the same for column space and left null space but in R^n.
But this stuff in the middle… I have no idea. I reread theory several times but didn't find a thing.
P.s. These pictures from "Introduction to Linear Algebra ed.5, Gilbert Strang".
Best Answer
The middle stuff in the top image is saying where $Ax$ lies if $x$ is a nonzero element of the row space (it goes to a nonzero element of the column space) or if $x$ is in the null space (it goes to the zero vector).
The middle stuff in the bottom image is showing what's in the top image, as well as what happens when $x$ is not in either the row space or null space (the generic situation). In this case one can write it as $x=x_r+x_n$ where $x_r$ is in the row space and $x_n$ is in the null space, and $Ax$ will be $A(x_r+x_n)=Ax_r+Ax_n=Ax_r$.