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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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  • Example of a linear transformation on R^3: rotation#23892
  • Example of a sum of vectors interpreted geometrically in R^2#23796
  • Example of (echelon matrix/matrix in (row) echelon form)#23541
  • Example of finding the inverse of a 2-by-2 matrix by row reducing the augmented matrix#23528
  • Example of finding the inverse of a 2-by-2 matrix by using a formula#23529
  • Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix#23530
  • Example of finding the inverse of a 3-by-3 matrix by using Cramer's rule#23531
  • Example of linear combination of vectors in R^2#23801
  • Example of matrix-vector product#23485
  • Example of multiplying 2x2 matrices#23492
  • Example of multiplying 3x3 matrices#23493
  • Example of multiplying matrices#23495
  • Example of multiplying nonsquare matrices#23494
  • Example of putting a matrix in echelon form#23537
  • Example of putting a matrix in echelon form and identifying the pivot columns#23538
  • Example of row reducing a 3-by-3 matrix#23546
  • Example of row reducing a 4-by-4 matrix#23547
  • Example of solving a 3-by-3 homogeneous matrix equation#23766
  • Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix#23762
  • Example of solving a 3-by-3 matrix equation#23765
  • Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix#23761
  • Example of using the echelon form to determine if a linear system is consistent.#23772
  • Example of vector-scalar multiplication in R^2#23794
  • Example of writing a given vector in R^3 as a linear combination of given vectors#23803
  • Examples#23886
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