📂Linear Algebra

Projection Theorem in Linear Algebra

Theorem¹

If $W$ is a subspace of a finite-dimensional inner product space $V$, then every $\mathbf{u} \in V$ is uniquely represented by the following formula.

$$\begin{equation} \mathbf{u} = \mathbf{w}_{1} + \mathbf{w}_{2} \end{equation}$$

Here, $\mathbf{w}_{1} \in W$ and $\mathbf{w}_{2} \in W^{\perp}$ apply.

Explanation

The notations $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ in the theorem are also marked as follows.

$$\mathbf{w}_{1} = \mathrm{proj}_{W} \mathbf{u} \quad \text{and} \quad \mathbf{w}_{2} = \mathrm{proj}_{W^{\perp}} \mathbf{u}$$

Moreover, these $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ are referred to as the orthogonal projection of $\mathbf{u}$ on $W$, orthogonal projection of $\mathbf{u}$ on $W^{\perp}$, respectively. $(1)$ can also be expressed as follows.

$$\mathbf{u} = \mathrm{proj}_{W} \mathbf{u} + \mathrm{proj}_{W^{\perp}} \mathbf{u}$$

$$\mathbf{u} = \mathrm{proj}_{W} \mathbf{u} + (\mathbf{u} - \mathrm{proj}_{W} \mathbf{u})$$