Why null space and column space? I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my mind forever, I really want to know what the purposes are for null space and column space of a vector.
Thanks!
 A: Perhaps an example will clarify things.
Let's suppose that the matrix A represents a physical system.  As an example, let's assume our system is a rocket, and A is a matrix representing the directions we can go based on our thrusters. So what do the null space and the column space represent?
Well let's suppose we have a direction that we're interested in.  Is it in our column space?  If so, then we can move in that direction.  The column space is the set of directions that we can achieve based on our thrusters.  Let's suppose that we have three thrusters equally spaced around our rocket.  If they're all perfectly functional then we can move in any direction.  In this case our column space is the entire range.  But what happens when a thruster breaks?  Now we've only got two thrusters.  Our linear system will have changed (the matrix A will be different), and our column space will be reduced.
What's the null space?  The null space are the set of thruster intructions that completely waste fuel.  They're the set of instructions where our thrusters will thrust, but the direction will not be changed at all.
Another example:  Perhaps A can represent a rate of return on investments.  The range are all the rates of return that are achievable.  The null space are all the  investments that can be made that wouldn't change the rate of return at all.
Another example:  room illumination.  The range of A represents the area of the room that can be illuminated.  The null space of A represents the power we can apply to lamps that don't change the illumination in the room at all.
Good luck!
