I'm stuck on a review problem.
Consider the matrix:
$\left[ \begin{array}{ccc} -1 & 1 \\ 1 & 1\\ 2 & 1 \end{array} \right] $
I'm asked to find a matrix $P$ which projects onto the range of $A$, with respect to the standard basis.
I'm not actually sure how to tackle this problem. I know that if you want to project a vector $u$ onto a vector $v$, you do $\operatorname{proj}_v u = \frac{\langle u,v\rangle}{\langle v.v\rangle}v$. Does a similar procedure exist if you want to project a matrix onto another matrix?
Also, what do they mean by range of $A$? By googling, it appears to refer to the column space, but if that's true, then what is the domain of a matrix? Does it even exist?
So it's easy to see that the two columns of $A$ are linearly independent, so they form a basis for $C(A)$. How should I go about solving the given problem? Also, what if I have to do it with respect to a non-standard basis?
Thanks for your help.