Orthogonality

Precise definition

Vectors $u,v$ are orthogonal when $\langle u,v\rangle=0$. A set is orthogonal when distinct vectors are pairwise orthogonal, and orthonormal when additionally every vector has norm 1. Any orthogonal set of nonzero vectors is linearly independent.

Notation and mathematical language

For a subspace $W$, $W^\perp=\{v:\langle v,w\rangle=0\text{ for all }w\in W\}$. In $\mathbb R^n$, $\operatorname{Row}(A)^\perp=\ker A$ and $\operatorname{Col}(A)^\perp=\ker A^T$.

Conceptual picture

Orthogonal directions do not interfere in the inner product. Pythagoras gives $\|u+v\|^2=\|u\|^2+\|v\|^2$ when $u\perp v$. Orthonormal coordinates satisfy $c_i=\langle v,q_i\rangle$.

Conditions and key results

Orthogonality depends on the inner product; two vectors may be orthogonal under one weighting but not another. Zero is orthogonal to every vector but cannot be part of an orthonormal set and destroys the standard independence conclusion if included.

A reliable strategy

1. State the inner product, including any weights or complex conjugation.
2. Compute all required pairwise inner products and norms.
3. Normalize nonzero orthogonal vectors if an orthonormal set is requested.
4. Use complement equations or coordinate formulas, then verify dimension and Pythagorean identities.

Fully worked example

Interpretation and application

Orthogonality simplifies energy, Fourier analysis, regression, error decomposition, and numerical algorithms. Unit choices affect a raw dot product, so applications may need nondimensionalization or a physically meaningful weighted inner product.

Common mistakes

Verification and reasonableness

• Recompute dot products in both orders and norms as nonnegative scalars.
• Check an orthogonal decomposition by summing components and testing residual inner products.
• Use dimension $\dim W+\dim W^\perp=n$ in finite Euclidean space.

Practice

1. Are $(1,1)$ and $(1,-1)$ orthogonal?
2. Does orthogonal imply independent?
3. What is $\ker A$ orthogonal to?

Further deduction

Bessel's inequality says that for an orthonormal set $q_1,\ldots,q_k$, $\sum|\langle v,q_i\rangle|^2\le\|v\|^2$. Equality holds when $v$ lies in their span, or for a complete orthonormal basis. It quantifies how much squared length the selected orthogonal components capture.

An orthogonal matrix $Q$ has $Q^TQ=I$, so $\|Qx\|=\|x\|$ and $(Qx)\cdot(Qy)=x\cdot y$. Its columns form an orthonormal basis and its determinant is $\pm1$. Rotations and reflections are the geometric examples; length preservation is the decisive algebraic property.

For complex vectors the inner product must conjugate one argument: $\langle x,y\rangle=\sum_i\overline{x_i}y_i$ under one common convention. Without conjugation, $\langle x,x\rangle$ need not be a nonnegative real number, so it cannot define length. Orthogonality still means zero inner product, and the Pythagorean identity follows for orthogonal vectors. Stating whether an inner product is linear in its first or second argument avoids apparent sign and conjugation disagreements between texts.

Every finite-dimensional inner-product space decomposes as $V=U\oplus U^\perp$. The two components are unique, their squared norms add, and the $U$ component is the closest vector in $U$. This theorem unifies projection, least squares, and Fourier coefficients.

Related topics

• Orthogonal Projection
• Gram-Schmidt Process
• Least Squares