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Multiplication by a change of coordinates matrix converts representations for different bases.
Change of coordinates matrices are invertible
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
Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.

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The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).

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Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.Close match
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).Close match
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).Close match
Definition of coordinate vector/mapping/representation relative to a given basisClose match
Coordinate vector spacesClose match
CoordinatesClose match
Definition of coordinates relative to a given basisClose match
BasesClose match
The standard/natural basis of R^n (or C^n) is a basis.Close match
Definition of the standard/natural basis of R^n (or C^n)Close match
Definition of basis of a vector space (or subspace)Close match
Each vector can be written uniquely as a linear combination of vectors from a given basis.Close match
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.Close match
Basis exchange using a nonzero coordinateClose match
Isomorphic vector spaces have the same dimension.Close match
A linear transformation is determined by its action on a basis.Close match
Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.Close match
The change of coordinates matrix between two bases exists and is uniqueClose match
Multiplication by a change of coordinates matrix converts representations for different bases.Close match
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)Close match

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