Orthogonal Complements

Definition of Orthogonal Complements

• Given a subspace $W$ of a vector space $V$, the orthogonal complement of $W$, denoted as $W^{\perp}$, includes all vectors in $V$ that are orthogonal to every vector in $W$.
• $W^{\perp} = {\mathbf{v} \in V : \mathbf{v} \cdot \mathbf{w} = 0 \text{ for all } \mathbf{w} \in W }$

Example 1 of Orthogonal Complements

• If $W = {\mathbf{0}}$, then $W^{\perp} = V$.

Example 2 of Orthogonal Complements

• If $W = V$, then $W^{\perp} = {\mathbf{0}}$.

Example 3 of Orthogonal Complements

• If $W = \text{span} \left{ \begin{bmatrix} 1 \ 0 \end{bmatrix} \right}$, then $W^{\perp} = \text{span} \left{ \begin{bmatrix} 0 \ 1 \end{bmatrix} \right}$.

Theorem of Orthogonal Complements

If $W$ is a subspace of $V$:

• $W^{\perp}$ is a subspace of $V$.
• $W \cap W^{\perp} = {\mathbf{0}}$.

Proof of Theorem Point 1

• Because $\mathbf{0} \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in W$, then $\mathbf{0} \in W^{\perp}$, so $W^{\perp}$ is nonempty.
• If $\mathbf{u}, \mathbf{v} \in W^{\perp}$, then for any $\mathbf{w} \in W$, $\mathbf{u} \cdot \mathbf{w} = 0$ and $\mathbf{v} \cdot \mathbf{w} = 0$. Therefore, $(\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w} = 0 + 0 = 0$, indicating $\mathbf{u} + \mathbf{v} \in W^{\perp}$.
• If $\mathbf{u} \in W^{\perp}$ and $c \in \mathbb{R}$, then for any $\mathbf{w} \in W$, $\mathbf{u} \cdot \mathbf{w} = 0$. Thus, $(c\mathbf{u}) \cdot \mathbf{w} = c (\mathbf{u} \cdot \mathbf{w}) = c(0) = 0$, implying $c\mathbf{u} \in W^{\perp}$.
• $W^{\perp}$ is a subspace of $V$.

Proof of Theorem Point 2

• Assume $\mathbf{v} \in W \cap W^{\perp}$.
• Because $\mathbf{v} \in W^{\perp}$, $\mathbf{v} \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in W$.
• Because $\mathbf{v} \in W$, $\mathbf{v} \cdot \mathbf{v} = 0$.
• $|\mathbf{v}|^2 = 0$, so $\mathbf{v} = \mathbf{0}$.

Relating Null Space to Row Space

• For an $m \times n$ matrix $A$, $(\text{row } A)^{\perp} = \text{Nul }A$

Proof of Theorem

• $\mathbf{x} \in (\text{row } A)^{\perp} \iff \mathbf{x}$ is orthogonal to every vector in row $A$
• $\iff A\mathbf{x} = \mathbf{0}$
• $\iff \mathbf{x} \in \text{Nul }A$

Corollary of Theorem

• $(\text{col }A)^{\perp} = \text{Nul }A^T$

Theorem of Double Orthogonal Complement

• For a subspace $W$ of $\mathbb{R}^n$, $(\text{W}^{\perp})^{\perp} = W$