Linear Transformation: Orthogonal Projections 
Define $\mathbf{u_1} =$ $\begin{align}    \begin{bmatrix}
            0 \\
            0 \\ 1 \\1\\
          \end{bmatrix}\end{align}$ and $\mathbf{u_2} =$ $\begin{align}    \begin{bmatrix}
            1 \\
            1 \\ 0 \\0\\
          \end{bmatrix}\end{align}$. Define $\mathbf{W} = \textrm{Span} \{\mathbf{u_1},\mathbf{u_2}\}$. Define two other vectors
   $\mathbf{u_3}$ and $\mathbf{u_4}$, such that the set $\mathbf{F} =
 \{\mathbf{u_1},\mathbf{u_2},\mathbf{u_3},\mathbf{u_4}\}$ forms an
   orthogonal basis for $\mathbb{R}^4$.
Let $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be the linear
   transformation which is orthogonal projection onto $\mathbf{W}$.
  
  
*
  
*What is $T(\mathbf{u_1})$ and $T(\mathbf{u_3})$?
  
*Find the matrix $M_T$ which represents the transformation $T$ with respect to the $\mathbf{F}$ coordinates.
  

Hi there, I am preparing for my linear algebra final examination. I stumbled across this question, however, no solutions were provided for this question. I am worried that I am studying the wrong thing. 
My approach to the questions:
First, I computed the vectors $\mathbf{u_3}$ and $\mathbf{u_4}$. After finding the null space for $\begin{align}    \begin{bmatrix}
            1 & 1 & 0 & 0\\
            0 & 0 & 1 & 1\\
          \end{bmatrix}\end{align}$, I obtained that $\mathbf{u_3} =$ $\begin{align}    \begin{bmatrix}
            -1 \\
            1 \\ 0 \\0\\
          \end{bmatrix}\end{align}$ and $\mathbf{u_4} =$ $\begin{align}    \begin{bmatrix}
            0 \\
            0 \\ -1 \\1\\
          \end{bmatrix}\end{align}$.


*

*$T(\mathbf{u_1}) = \frac {\mathbf{u_1} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}}\mathbf{u_1} +  \frac {\mathbf{u_1} \cdot \mathbf{u_2}}{\mathbf{u_2} \cdot \mathbf{u_2}}\mathbf{u_2} = \mathbf{u_1}$. 

*$T(\mathbf{u_3}) = \frac {\mathbf{u_3} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}}\mathbf{u_1} +  \frac {\mathbf{u_3} \cdot \mathbf{u_2}}{\mathbf{u_2} \cdot \mathbf{u_2}}\mathbf{u_2} = \mathbf{0}$ (since $\mathbf{u_3}$ is orthogonal to both $\mathbf{u_1}$ and $\mathbf{u_2}$)

*We are working on the $\mathbf{F}$ coordinates. Note that $\mathbf{u_1}, \mathbf{u_2} \in \mathbf{F}$ and $\mathbf{u_3}, \mathbf{u_4} \in \mathbf{F^\perp}$. Thus, each basis vectors is an eigenvector for the projection. Hence, the matrix $M_T$ is given by:
$$\begin{align} \begin{bmatrix} 1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{bmatrix} \end{align}$$


Are my solutions and reasonings correct? Any insight on these would be highly appreciated. Thank you!

I think that your solution is correct, other than the fact that you computed $\mathbf{u}_3$ and $\mathbf{u}_4$. Note that the completion of an orthogonal linearly independent set to an orthogonal base is not unique; however, the rest of the question can be answered even without knowing who the $\mathbf{u}_i$-s are.
