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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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526–550 of 715 nodes
  • Proof of several equivalences for nonsingular matrix#23644
  • QR decomposition#23630
  • Rank and mullity#23672
  • Rank and nullity#23882
  • Removing a linearly dependent vector from a set does not change the span of the set.#23982
  • R^n is a vector space.#24097
  • Row equivalence is an equivalence relation#23480
  • Row equivalent matrices have the same row space.#23881
  • Row equivalent matrices represent equivalent linear systems#23769
  • Row operations#23474
  • Row operations are given by multiplication by elementary matrices.#23559
  • Row operations do not necessarily preserve the column space.#23878
  • Scalar multiplication#23470
  • Similarity of matrices#23634
  • Similarity of matrices in an equivalence relation.#23636
  • Similar matrices have the same eigenvalues and the same characteristic polynomials.#23638
  • Spans#23804
  • Subspaces#23811
  • Subspaces associated to a linear transformation#24055
  • Subspaces associated to a matrix#23872
  • Switching two rows multiplies the determinant by -1.#23738
  • Symmetric matrices#23571
  • Symmetric matrices are square.#23573
  • Terminology#23754
  • The 0 scalar multiplied by any vector equals the 0 vector.#23960
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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