13. Orthogonality

Two vectors are orthogonal (written ) if their dot product is zero:

In and , this is the same as perpendicular. In higher dimensions or abstract inner product spaces, orthogonality generalizes the notion.

13.1. Orthogonal sets and orthonormal sets

A set is:

• Orthogonal if every pair is orthogonal: for
• Orthonormal if also each is a unit vector:

A useful concise formula uses the Kronecker delta:

13.2. Key facts

• Linear independence: any orthogonal set of non-zero vectors is automatically linearly independent
• Pythagoras: if , then
• Easy coordinates: in an orthonormal basis , the coordinates of are just dot products:

13.3. Orthogonal complement

For a subspace , its orthogonal complement is:

— the set of vectors orthogonal to every vector in .

Properties:

• is itself a subspace
• (every vector decomposes uniquely as )

13.4. Four fundamental subspaces (and their complements)

For an matrix , four subspaces and two pairings:

The column space is orthogonal to the left null space; the row space (= column space of ) is orthogonal to the null space.

13.5. Orthogonal projection

For a subspace with orthonormal basis , the projection of onto is:

In matrix form with : . The matrix is the projection matrix — see Projection.

13.6. Why orthogonality matters

• Best approximation: projecting onto a subspace gives the closest point — basis of least-squares fitting
• Decoupling: in an orthonormal basis, coordinates are independent — no cross-terms, simpler computations
• Orthogonal matrices preserve distances and angles — rigid transformations
• Spectral theorem: real symmetric matrices have orthogonal eigenvectors
• Gram–Schmidt builds orthonormal bases from any basis

13.7. See also

• Dot Product
• Projection
• Gram–Schmidt
• Orthogonal Matrix
• Inner Product — generalization