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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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Pythagorean equality characterizes orthogonality over the reals

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OrthogonalityClose match
The triangle inequalityClose match
Norm and lengthClose match
Definition of orthogonal vectorsClose match
Inner products in coordinate spacesClose match
The Cauchy-Schwarz inequalityClose match
The Cauchy–Schwarz inequalityClose match
A matrix with real entries and orthonormal columns preserves dot products.Close match
The standard inner product of a vector with itself is non-negativeClose match
The standard inner product on R^n is commutative.Close match
The standard inner product on R^n commutes with (real) scalar multiplication.Close match
A matrix with real entries and orthonormal columns preserves norms.Close match
The inner product of a vector with itself is the square of its norm/length.Close match
The standard inner product of a vector with itself is 0 only for the 0 vectorClose match
Complex inner products conjugate scalars in the first argumentRelated
The orthogonal complement of a subspace is a subspace.Related
Definition of orthonormal set of vectorsRelated
Definition of parallel vectorsRelated
Definition of angle between vectorsRelated
Definition of orthonormal basis of a (sub)spaceRelated

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