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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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Multiplication by a change of coordinates matrix converts representations for different bases.

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Definition of change of coordinates matrix between two basesClose match
Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.Close match
CoordinatesClose match
The change of coordinates matrix between two bases exists and is uniqueClose match
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)Close match
Definition of coordinates relative to a given basisClose match
A linear transformation is given by a matrix with respect to a given basis.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
Change of coordinates matrices are invertibleClose match
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spacesClose match
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).Close match
BasesClose match
Basis exchange using a nonzero coordinateClose match
Example of multiplying 2x2 matricesClose match
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).Close match
Example of multiplying 3x3 matricesRelated
Matrix multiplication is associative.Related
Definition of matrix multiplicationRelated
Example of multiplying matricesRelated

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