Projection Transformation

$\mathrm{Proj}(y;A)={\text{argmin}}_{v\in \mathcal{R}(A)}\Vert v-y{\Vert }_2=\mathbf{A}({\mathbf{A}}^{\top }\mathbf{A}{)}^{-1}{\mathbf{A}}^{\top }y$

• Projection of a vector $y$ onto the a matrix $A$‘s columnspace

$\text{Proj}(y;a)=\frac{a{a}^{\top }}{a^{\top }a}y$

• Projection of a vector $y$ onto another vector $a$