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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.
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Each vector can be written uniquely as a linear combination of vectors from a given basis.

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A set is a basis if each vector can be written uniquely as a linear combination.Close match
BasesClose match
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).Close match
Definition of basis of a vector space (or subspace)Close match
Definition of coordinates relative to a given basisClose match
CoordinatesClose match
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).Close match
Basis exchange using a nonzero coordinateClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
A vector can be written uniquely as a linear combination of vectors from independent subspaces.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
The union of bases from independent subspaces is a basis for the space.Close match
A set of nonzero vectors contains (as a subset) a basis for its span.Close match
The standard/natural basis of R^n (or C^n) is a basis.Close match
If a set of vectors contains the 0 vectorClose match
Definition of coordinate vector/mapping/representation relative to a given basisClose match
Every basis for a vector space contains the same number of elementsClose match
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).Close match
The change of coordinates matrix between two bases exists and is uniqueClose match
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.Close match

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