Projection, Linear transformation, maybe change of basis.

This is my problem.

Let V=Span([1,0,2,1],[0,1,-1,1]) $\displaystyle \subset R^4 $. Find the standard matrix for the linear transformation $\displaystyle proj_{V}:R^4 \rightarrow R^4 $.

Here is my thinking.

Setting the matrix $\displaystyle P=A(A^\top A)^{-1} A^\top where A= \left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right] $. This would be the transformation matrix with respect to the given basis of V. Normally, the standard matrix T would be $\displaystyle T=APA^{-1} $ but A cannot be $\displaystyle \left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right] $ because that is not a square matrix and obviously not invertible. What do I do?

Maximum explanation please because I am still trying to wrap my head around these concepts. Thanks!!!