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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 matrix with real entries and orthonormal columns preserves dot products.

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A matrix with real entries and orthonormal columns preserves norms.Close match
A square real matrix has orthonormal columns iff its inverse is its transposeClose match
Unitary matrices preserve inner products.Close match
Definition of orthonormal set of vectorsClose match
Definition of orthogonal matrixClose match
Inner products in coordinate spacesClose match
Hermitian matricesClose 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
Inner productsClose match
Pythagorean equality characterizes orthogonality over the realsClose match
The complex inner product is conjugate symmetricClose match
The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.Close match
A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.Close match
Norm and lengthRelated
OrthogonalityRelated
If two matrices have equal products with all vectorsRelated
Matrix-vector product is associativeRelated
Definition of orthogonal vectorsRelated
Example of matrix-vector productRelated

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