# Orthogonal projection: Find vector $\overrightarrow w \in \Bbb R^2$ so that $\overrightarrow w$ is orthogonal to $\overrightarrow v$

Problem: Let $$l=L \begin{bmatrix} 3 \\ 4 \\ \end{bmatrix}$$ be line in $\Bbb R^2$ and $\mathcal A :\Bbb R^2 \rightarrow \Bbb R^2$ orthogonal projection onto line $l$. Find $\mathcal A$ (standard basis) and find vector $\overrightarrow w \in \Bbb R^2$ so that $\overrightarrow w$ is orthogonal to $$\overrightarrow v = \begin{bmatrix} 3 \\ 4 \\ \end{bmatrix}$$ and then find matrix of that linear operator using basis that is composed of vectors $\overrightarrow w$ and $\overrightarrow v$.

I have no idea how to do this. I was told to form standard base, so I tried making a new base $B=\{(3,4), (1,0)\}$. Then I assume there exists some vector $(x,y) \in \Bbb R^2$ such that $$\mathcal A(x,y)=\alpha (3,4) + \beta (1,0)$$ I solve this and find $\alpha=\frac{y}{4}$. Is my linear operator now: $$\mathcal A(x,y)=\frac{y}{4}(3,4)=(\frac{3y}{4},y)?$$

Now, instead of $(x,y)$ I write $(1,0)$ and $(0,1)$ and I have my matrix in standard base. Is this correct?

I am not sure how to find second part of the problem. If someone could explain, I would be really greatful. Thank you.

• Hint : use basic propriety of inner product – Hamza Jul 10 '16 at 23:55
• Also \vec{} is a quicker way to write letters with arrows over them like $\vec{x}$ for future reference – Triatticus Jul 11 '16 at 0:07

The matrix for $\mathcal A$ with respect to the standard basis {$e_1, e_2$} is going to be the matrix whose two columns are $Ae_1$ and $Ae_2$. You can compute what each of those vectors should be using the linear transformation itself.
For the second part of your question, you should use the property of the dot product in $\mathbb R^2$ that two orthogonal vectors have $v\cdot w = 0$. That being said, once you have determined the vector $w$, the matrix of $\mathcal A_\beta$ with respect to the basis $\beta = \{v, w\}$ takes on the particularly simple form:
$\begin{pmatrix} \mathcal {A}_\beta v_\beta & \mathcal A_\beta w_\beta \end{pmatrix}.$
With respect to $\beta$, what should $v_\beta$ and $w_\beta$ be?