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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
  • Basis exchange using a nonzero coordinate
  • 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. Coordinates

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701–718 of 718 nodes
  • Transpose and adjoint#23502
  • Transpose commutes with scalar multiplication.#23511
  • Triangularization requires the characteristic polynomial to split#24076
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are dependent iff at least one is a scalar multiple of the other#23826
  • 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 form of solutions to a consistent linear system#23444
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • 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
  • Container for Linear Algebra#24123
  • Math#23168
  • Test Flight Problem Set#23151

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