Exercise 7

We have three ways to find the orthogonal projection of a vector into a line, the Definition 1.1 way from the first subsection of this section, the Example 3.2 and 3.3 way of representing the vector with respect to a basis for the space and then keeping the $M$ part, and the way of Theorem 3.8. For these cases, do all three ways.

• $\vec{v}=\begin{pmatrix}1\\ -3\end{pmatrix},\quad M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\mid x+y=0\hskip 1.00006pt\}$}$

• $\vec{v}=\begin{pmatrix}0\\ 1\\ 2\end{pmatrix},\quad M=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid\text{$x+z=0$ and $y=0$}\hskip 1.00006pt\}$}$