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Coordinate vector spaces
Abstract vector spaces

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Algebraic properties of R^n (or C^n)
Axioms of a vector space
Spans
Linear (in)dependence
Bases
Dimension
Linear transformations
Definition and terminology
Linear combinations
Subspaces
Basic properties
Isomorphism
Geometric properties of R^n (or C^n)
Orthogonality and projection

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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 basis of a vector space (or subspace)Close match
Definition of coordinates relative to a given basisClose match
CoordinatesClose match
Each vector can be written uniquely as a linear combination of vectors from 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
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).Close match
The standard/natural basis of R^n (or C^n) is a basis.Close match
DimensionClose match
A set is a basis if each vector can be written uniquely as a linear combination.Close match
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).Close match
Vector spacesClose match
SpansClose match
Definition of coordinate vector/mapping/representation relative to a given basisClose match
Linear combinationsClose match
Coordinate vector spacesClose match
Every basis for a vector space contains the same number of elementsClose match
Definition of the standard/natural basis of R^n (or C^n)Close match
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.Close match
Basis exchange using a nonzero coordinateClose match
Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.Close match

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