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  • Bases

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  • Definition of basis of a vector space (or subspace)
  • Definition of the standard/natural basis of R^n (or C^n)
  • The standard/natural basis of R^n (or C^n) is a basis.
  • Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
  • Definition of the standard basis of the polynomials of degree at most n
  • Definition of the standard basis of the m by n matrices
  • Definition of coordinates relative to a given basis
  • A set of nonzero vectors contains (as a subset) a basis for its span.
  • The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
  • Each vector can be written uniquely as a linear combination of vectors from a given basis.
  • A set is a basis if each vector can be written uniquely as a linear combination.
  • Coordinates

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  • Definition of coordinates relative to a given basis
  • If B is a basis containing b and the b coordinate of c is nonzero
  • Definition of coordinate vector/mapping/representation relative to a given basis
  • The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).
  • The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).
  • The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).
  • Definition of change of coordinates matrix between two bases
  • The change of coordinates matrix between two bases exists and is unique
  • Multiplication by a change of coordinates matrix converts representations for different bases.
  • Change of coordinates matrices are invertible
  • Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.
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451–475 of 715 nodes
  • Geometric properties of R^n (or C^n)#23795
  • Hermitian matrices#23596
  • Hermitian matrices have real eigenvalues.#23711
  • Homogeneous linear systems are consistent.#23434
  • If A and B are n-by-n matrices#23729
  • If A is a matrix#23679
  • If a matrix has both a left and a right inverse#23519
  • If a set of vectors contains the 0 vector#23825
  • If a set of vectors in R^n (or C^n) contains more than n elements#23827
  • If a space is the direct sum of invariant subspaces#24053
  • If a square matrix has a one-sided inverse#23520
  • If a vector space has dimension n#23838
  • If B is a basis containing b and the b coordinate of c is nonzero#23990
  • If the product of a vector and a scalar is 0#23962
  • If two finite dimensional subspaces have the same dimension and one is contained in the other#24011
  • If two matrices have equal products with all vectors#23484
  • Inner products#24087
  • Inner products in coordinate spaces#23894
  • Inverse#23514
  • Isomorphic vector spaces have the same dimension.#24004
  • Isomorphism#24000
  • Least squares#23945
  • Linear algebra#23410
  • Linear combinations#23799
  • Linear (in)dependence#23815
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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