Consider the [[Orthogonal Projection in Real n-Space|orthogonal projection]] from vectors in $\mathbb{R}^{2}$ onto the [[Lines and planes in Real 3-Space#^393c09|line]] $\ell=(2x-3y=0)\subset\mathbb{R}^2.$ We apply the formula: let $\hat{\underline{w}}$ be the unit vector along the line $\ell$ in the and then map $\underline{v} \mapsto (\underline{v}\cdot \underline{\hat{w}})\hat{\underline{w}}$. The vector $\underline{w}=(3,2)^{T}$ lies on $\ell$ and the unit vector in that direction is $\hat{\underline{w}} = \frac{1}{\sqrt{ 13 }} \begin{pmatrix} 3 \\ 2 \end{pmatrix}$so the [[Linear maps|linear map]] is $\left.\begin{pmatrix}x\\y\end{pmatrix}\mapsto\left(\begin{pmatrix}x\\y\end{pmatrix}\right.\cdot\underline{\hat{w}}\right)\underline{\hat{w}}=\frac1{13}(3x+2y)\begin{pmatrix}3\\2\end{pmatrix}=\frac{1}{13} \begin{pmatrix} 9x + 4y \\ 6x+4y \end{pmatrix}A\begin{pmatrix}x\\y\end{pmatrix}$where $A=\frac1{13}\begin{pmatrix}9&6\\6&4\end{pmatrix}.$ In other words, the orthogonal projection to the line $\ell$ is the [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] $L_{A}:\mathbb{R}^{2}\to \mathbb{R}^{2}$ for the matrix $A$. Its *image* is given by $\text{Im }A= \text{Colspan } A = \langle (9,6)^{T}, (6,4)^{T} \rangle = (3,2)^{T} \mathbb{R} = \ell$. Its *kernel* is given by $\ker A = \ker \begin{pmatrix} 3 & 2 \\ 0 & 0 \end{pmatrix} = (3x+2y = 0) \subset \mathbb{R}^{2}$which is the line $\ell^{\perp} =(3x+2y=0)$ which is indeed at right angles to $\ell$. Furthermore, for any point $P=(a,b)^{T}\in\ell$, the set of points that maps to $P$ is simply $P+\ell^{\perp}$. ![[Orthogonal projection of vector to line in a plane.png|400]]