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Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.
Definition of isomorphic/isomorphism between vector spaces
The inverse of an isomorphism is an isomorphism.
Vector space isomorphism is an equivalence relation.
Isomorphic vector spaces have the same dimension.
Finite-dimensional vector spaces of equal dimension over the same field are isomorphic
Definition of automorphism of a vector space

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Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.

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The coordinate vector relative to a given basis is a surjective 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
DimensionClose match
Isomorphic vector spaces have the same dimension.Close match
Definition of coordinate vector/mapping/representation relative to a given basisClose match
Coordinate vector spacesClose match
BasesClose match
The standard/natural basis of R^n (or C^n) is a basis.Close match
Finite-dimensional vector spaces of equal dimension over the same field are isomorphicClose match
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).Close match
Definition of isomorphic/isomorphism between vector spacesClose match
IsomorphismClose match
Definition of the standard/natural basis of R^n (or C^n)Close match
Every basis for a vector space contains the same number of elementsClose match
Definition of automorphism of a vector spaceClose match
R^n is a vector space.Close match
If a vector space has dimension nClose match
Definition of R^n (or C^n)Close match
Algebraic properties of R^n (or C^n)Close match
If a set of vectors in R^n (or C^n) contains more than n elementsClose match

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