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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

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Definition of the standard basis of the polynomials of degree at most n

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The set of all polynomials of degree at most n is a vector space.Close match
Definition of the standard/natural basis of R^n (or C^n)Close match
Definition of the standard basis of the m by n matricesClose match
The standard/natural basis of R^n (or C^n) is a basis.Close match
BasesClose match
The set of all polynomials is a vector space.Close match
Definition of basis of a vector space (or subspace)Related
Each vector can be written uniquely as a linear combination of vectors from a given basis.Related
Definition of coordinates relative to a given basisRelated
DimensionRelated
CoordinatesRelated
If a vector space has dimension nRelated
A set is a basis if each vector can be written uniquely as a linear combination.Related
Basis exchange using a nonzero coordinateRelated
Definition of R^n (or C^n)Related
Definition of coordinate vector/mapping/representation relative to a given basisRelated
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)Related
Every basis for a vector space contains the same number of elementsRelated
Definition of orthonormal basis of a (sub)spaceRelated
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).Related

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