TITLE: Visualizing the four subspaces of a matrix
QUESTION [7 upvotes]: Given a system of linear equations in the form $$AX=b$$ How can I go about visualizing the four fundamental sub-spaces - column space, row space, null space and left null space?
In the same context, how can I visualize the orthogonality of row space and null space, and column space and the left null space?

REPLY [3 votes]: I could type it all out, but I think this most efficiently gets you toward what you are after. 

Here is the original source.

Here is another way to think about these things from Gilbert Strang: