Projection Theorem

Theorem

If $W$ is a finite-dimensional subspace of an inner product space $V$, then every vector $\mathbf{u}$ in $V can be expressed in exactly one way as

$$\mathbf{u}={\mathbf{w}}_1+{\mathbf{w}}_2$$

where ${\mathbf{w}}_1\in W$ and ${\mathbf{w}}_2\in W^{\perp }$