Projection onto Column Space of A and its Perpendicular

[Zyxer22]

Homework Statement

Some of the details in this question are based off the use of matlab. If it's needed I can show the matrices that MATLAB creates.


Let A = magic(8); A = A(:,1:3) and let S be the Column Space of A. For b = [1:8]' compute the projection of b onto the Column Space of A. What is the projection of b perpendicular to the Column Space of A?

Homework Equations

The Attempt at a Solution

I'm not sure exactly where to get started. My first issue is that I'm not sure how to find the column space of A. I think I'm supposed to take the reduced row echelon form of the matrix and then shrink it if necessary. My book defines Column space as the span, so this would make sense to me.

I'm also not sure how to find the perpendicular of a matrix. I know it's defined such that v \in W^\perp => v \cdot u = 0 | u \in W but I'm not sure how to go from there.

 

[Zyxer22]

I'm fairly sure I'm right about the column space but I could use an explanation on perpendicular matrices if anyone can help.