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  • 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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  1. 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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  • Definition of Hessenberg form#23619
  • Definition of homogeneous linear system of equations#23433
  • Definition of how the action of a linear transformation on a basis extends to the whole space#23867
  • Definition of identity linear transformation#24014
  • Definition of identity matrix#23457
  • Definition of ill-conditioned linear system#23446
  • Definition of image (of a point) under a linear transformation#23842
  • Definition of inconsistent linear system#23432
  • Definition of independent subspaces#23972
  • Definition of index of nilpotency#23584
  • Definition of inner/dot product on C^n#23896
  • Definition of inner/dot product on R^n#23895
  • Definition of inner product#24088
  • Definition of inner product space#24089
  • Definition of intersection of subspaces#23966
  • Definition of invariant subspace of a linear transformation.#24052
  • Definition of inverse of a linear transformation#23849
  • Definition of inverse of a matrix#23517
  • Definition of invertible linear transformation#23848
  • Definition of invertible matrix#23518
  • Definition of invertible/nonsingular linear transformation#24021
  • Definition of isomorphic/isomorphism between vector spaces#24001
  • Definition of Jordan form#23620
  • Definition of kernel/null space of linear transformation#24058
  • Definition of kernel of linear transformation#23847
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