Projection

Definition

A projection on $V$ is an operator $E\in \mathcal{L}(V)$ such that $E^2=E$ A matrix $A$ is a projection if $A^2=A$

Properties

• With $R=range(E)$
• With $N=ker(E)$ Then,

1. $\beta \in R⟺\beta =E(\beta )$ (The vector is within the range, if and only if the projection of the vector is equal to itself)
2. $\forall \alpha \in V,E(\alpha )\in R\wedge (I-E)\alpha =N$
3. $E$ is Diagonalizable
4. $V=R\oplus N$, $\forall v\in V,v=E(v)+(I-E)(v)$
5. Nice matrix representation. ${\beta }_R$ as a basis for $R$, ${\beta }_N$ as a basis for $N$, then $[E{]}_{\{{\beta }_R,{\beta }_N}\}=\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]$

Concepts

• Finding a Projection Along a Subspace