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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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  • Examples of vector spaces#24095
  • Factorization of matrices#23622
  • F^n is a vector space.#24099
  • For invertible linear transformations A and B#24030
  • For matrices#23500
  • Formula for computing the least squares solution to a linear system#23949
  • Formula for computing the least squares solution to a linear system.#23948
  • Formula for diagonalizing a real 2-by-2 matrix with a complex eigenvalue.#23618
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.#23933
  • Formula for the coordinates of the projection of a vector onto a subspace#23944
  • Formula for the determinant of a 2-by-2 matrix.#23742
  • Formula for the determinant of a 3-by-3 matrix.#23743
  • Formula for the inverse of a 2-by-2 matrix.#23521
  • Formula for the least-squares linear fit to 2-dimensional data#23952
  • Formula for the (orthogonal) projection of one vector onto another vector#23939
  • Formula for the spectral decomposition for a symmetric matrix#23576
  • For n-by-n invertible matrices A and B#23523
  • Gaussian elimination as a method to solve a linear system#23421
  • Gauss-Jordan procedure to put a matrix into reduced row echelon form#23540
  • Geometric description of span of a set of vectors in R^n (or C^n)#23806
  • Geometric picture of a 2-by-2 linear system#23428
  • Geometric picture of a 3-by-3 linear system#23430
  • Geometric picture of the solution set of a linear equation in 3 unknowns#23429
  • Geometric properties of linear transformations#23851
  • Geometric properties of linear transformations on R^2#23853
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