# Thread: Projection, Linear transformation, maybe change of basis.

1. ## Projection, Linear transformation, maybe change of basis.

This is my problem.

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

Here is my thinking.
Setting the matrix $P=A(A^\top A)^{-1} A^\top where A= \left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right]$. This would be the transformation matrix with respect to the given basis of V. Normally, the standard matrix T would be $T=APA^{-1}$ but A cannot be $\left[\begin{array}{cc}1&0\\0&1\\2&-1\\1&1\end{array}\right]$ 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!!!

2. There haven't been any views yet so I doubt anyone cares but I figured it out. The question was in the section of my book called Change of Basis so I did the problem using that, only to find out that a lot cancels and the answer comes out to be the matrix P that I denote above. You have to find two vectors that span V perp so that is a new basis of R^4, but when you apply the linear transformation and the change of basis formula, they cancel out.