Linear Algebra
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Orthogonal Complement

Given a subspace VV of Rm\mathbb{R}^m, its orthogonal complement V⊥V^\perp is the set of all vectors in Rm\mathbb{R}^m that are orthogonal to every vector in VV. It is the collection of all "directions perpendicular to VV."

In R3\mathbb{R}^3, the complement of a line through the origin is a plane, and vice versa. The dimensions add up: dim⁡(V)+dim⁡(V⊥)=m\dim(V) + \dim(V^\perp) = m. And taking the complement twice returns the original: (V⊥)⊥=V(V^\perp)^\perp = V.

A vector n\mathbf{n} orthogonal to every vector in a hyperplane VV is called a normal vector to VV. If the hyperplane is defined by the equation ntx=b\mathbf{n}^t \mathbf{x} = b, then n\mathbf{n} is a normal vector. Normal vectors are the key to describing hyperplanes without choosing a basis.

Formal View

Definition 6.9 — Orthogonal Complement
Let VV be a subspace of Rm\mathbb{R}^m. Its orthogonal complement is
V⊥:={z∈Rm:z⋅v=0 for all v∈V}.V^\perp := \{\mathbf{z} \in \mathbb{R}^m : \mathbf{z} \cdot \mathbf{v} = 0 \text{ for all } \mathbf{v} \in V\}.
Theorem 6.10
Let VV be an nn-dimensional subspace of Rm\mathbb{R}^m with basis {a1,…,an}\{\mathbf{a}_1, \ldots, \mathbf{a}_n\}:\begin{enumerate} \item V⊥V^\perp is a subspace of Rm\mathbb{R}^m. \item V∩V⊥={0}V \cap V^\perp = \{\mathbf{0}\}. \item z∈V⊥\mathbf{z} \in V^\perp iff z⋅ai=0\mathbf{z} \cdot \mathbf{a}_i = 0 for each basis vector ai\mathbf{a}_i. \item dim⁡(V⊥)=m−n\dim(V^\perp) = m - n and (V⊥)⊥=V(V^\perp)^\perp = V. \end{enumerate}

Property 3 is computationally useful: you only need to check orthogonality against a basis, not every vector in V.

Why This Matters

Orthogonal complements decompose space into complementary pieces — a fundamental tool across all of applied mathematics.

  • The four fundamental subspaces of a matrix (column space, null space, row space, left null space) come in orthogonal complement pairs
  • In quantum mechanics, the state space decomposes as V⊕V⊥V \oplus V^\perp; measuring collapses a state to one of these pieces
  • Noise cancellation in audio works by projecting a signal onto the subspace orthogonal to known noise patterns

Quiz

Question 1

If VV is a 2-dimensional subspace of R5\mathbb{R}^5, what is dim⁡(V⊥)\dim(V^\perp)?

Question 2

If z\mathbf{z} is orthogonal to every basis vector of VV, then z∈V⊥\mathbf{z} \in V^\perp.

Common Mistakes

  • Thinking V⊥V^\perp is just one vector — the orthogonal complement is a full subspace, usually of positive dimension.
  • Forgetting V∩V⊥={0}V \cap V^\perp = \{\mathbf{0}\} — a nonzero vector cannot be orthogonal to itself (by positivity of the dot product).
  • Confusing orthogonal complement with the set complement — V⊥V^\perp is a subspace, not "everything outside VV."