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Orthogonality

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Definition of angle between vectors
Definition of orthogonal vectors
Definition of parallel vectors
Pythagorean equality characterizes orthogonality over the reals
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.
Null spaces are orthogonal complements of the corresponding row or column spaces
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 square real matrix has orthonormal columns iff its inverse is its 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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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.

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The orthogonal complement of a subspace is a subspace.Close match
Definition of a vector being orthogonal to a subspaceClose match
The direct sum of a subspace and its orthogonal complement is the whole space.Close match
Definition of orthogonal complement of a subspaceClose match
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.Close match
Definition of orthogonal basis of a (sub)spaceClose match
Definition of orthogonal subspacesClose match
The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.Close match
Definition of complement of a subspaceClose match
Orthogonality and projectionClose match
Definition of orthonormal basis of a (sub)spaceClose match
OrthogonalityClose match
Definition of orthogonal vectorsClose match
Definition of orthogonal set of vectorsClose match
Definition of basis of a vector space (or subspace)Close match
Formula for the coordinates of the projection of a vector onto a subspaceClose match
A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinationsClose match
Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.Close match
An orthogonal set of nonzero vectors is linearly independent.Close match
A vector can be written uniquely as a linear combination of vectors from independent subspaces.Close match

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