Problems about orthogonal projection

Question

I'm now encountering a problem about orthogonal projection and change of basis. I'm asked to find a transformation matrix for the linear transformation $proj_{W}(x)$ and $proj_{W^{\perp}}(x)$, where W is a space spanned by the set of vectors ${[1,2,3,-4]^{T}, [2,3,-4,1]^{T}, [2,-5,8,-3]^{T}, [5,26,-9,-12]^{T}, [3,-4,1,2]^{T}}$ and x is the vector $[1,0,-3,-7]^{T}$. I'm required to use the method changing the basis of vectors, that is $$ [T(u)]_{\beta v} = [T]_{\beta u \beta v}[u]_{\beta u} $$ I know that to solve this question, I first need to reduce it to the rref, which is $$ \begin{bmatrix} 1 & 0 & 0 & 5 & -1 \\ 0 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} $$, which means that the span space W's basis is the set $$ {[1,0,0,0]^{T}, [0,1,0,0]^{T}, [0,0,1,0]^{T}} $$, then I can find the [T] for transformation matrix is $$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} $$ Then the problem comes, using this method, I will find the answer to be $[1,0 ,-3]^{T}$, which is not equal to the answer got by directly computing the orthogonal projection, which is $[1,0,-3,0]^{T}$. I'm really confused, could you figure out the problems here? Thanks!