Orthogonality
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- Definition of angle between vectors
- Definition of orthogonal vectors
- Definition of parallel vectors
- Two vectors are orthogonal if and only if the Pythagorean Theorem holds.
- Definition of a vector being orthogonal to a subspace
- Definition of orthogonal complement of a subspace
- The orthogonal complement of a subspace is a subspace.
- The direct sum of a subspace and its orthogonal complement is the whole space.
- A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.
- The null space of a matrix is the orthogonal complement of the column space.
- Definition of orthogonal set of vectors
- Definition of orthonormal set of vectors
- An orthogonal set of nonzero vectors is linearly independent.
- Definition of orthogonal basis of a (sub)space
- Definition of orthonormal basis of a (sub)space
- A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.
- A matrix with real entries and orthonormal columns preserves norms.
- A matrix with real entries and orthonormal columns preserves dot products.
- Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
- A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
- Description of the Gram-Schmidt process
- The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
- Definition of Gram-Schmidt process
- Definition of orthogonal subspaces
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