Projection

In linear algebra, a projection decomposes a vector $\vec{x}$ into a combination of two: one ${\vec{x}}^{\Vert }$ that is parallel to some line $L$ in the plane, and one ${\vec{x}}^{\perp }$ that is perpendicular. The easiest formula is below, for the projection of $\vec{x}$ onto $\vec{u}$.

\text{proj}_\vec u(\vec x)=\bigg(\frac{\vec u \cdot \vec x}{\|\vec u\|^2}\bigg)\vec u

Theorems

• Suppose $V$ is a subspace of ${\mathbb{R}}^n$ and $\vec{x}\in {\mathbb{R}}^n$. The closest vector in $V$ to $\vec{x}$ is given by ${\text{Proj}}_V(\vec{x})$. In other words, $\Vert {\text{Proj}}_V(\vec{x})-\vec{x}\Vert \le \Vert \vec{v}-\vec{x}\Vert$ for any $\vec{v}\in V$.

See also

• Gram-Schmidt algorithm, which heavily uses projections