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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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  1. Definition of coordinate vector/mapping/representation relative to a given basis

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701–715 of 715 nodes
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.#23918
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector spaces with the same dimension are isomprphic.#24005
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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