# 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 $\mathbf{W}$.

1. What is $T(\mathbf{u_1})$ and $T(\mathbf{u_3})$?
2. 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.
• Was typing the very same point. It's not necessary to compute $\mathbf{u}_3$ and $\mathbf{u}_4$ to complete the problem. On an exam, where time is constrained, it's important to recognize that. – Matthew Leingang Nov 8 '15 at 10:32