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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
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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Definition of coordinates relative to a given basis
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