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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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Change of coordinates matrices are invertible
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