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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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Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.

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Definition of orthogonal basis of a (sub)spaceClose match
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).Close match
Definition of coordinates relative to a given basisClose match
Definition of orthonormal basis of a (sub)spaceClose match
BasesClose match
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).Close match
Definition of coordinate vector/mapping/representation relative to a given basisClose match
Formula for the coordinates of the projection of a vector onto a subspaceClose match
Formula for the (orthogonal) projection of one vector onto another vectorClose match
Orthogonality and projectionClose match
Definition of basis of a vector space (or subspace)Close match
Basis exchange using a nonzero coordinateClose match
Definition of (orthogonal) projection of one vector onto another vectorClose match
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).Close match
Definition of orthogonal vectorsClose match
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.Close match
OrthogonalityClose match
Definition of a vector being orthogonal to 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 set of vectorsClose match

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