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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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Isomorphic vector spaces have the same dimension.

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Finite-dimensional vector spaces of equal dimension over the same field are isomorphicClose match
Definition of isomorphic/isomorphism between vector spacesClose match
IsomorphismClose match
Every basis for a vector space contains the same number of elementsClose match
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.Close match
Vector space isomorphism is an equivalence relation.Close match
DimensionClose match
If a vector space has dimension nClose match
Definition of dimension of a vector space (or subspace)Close match
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).Close match
BasesClose match
Axioms of a vector spaceClose match
The dimension of a subspace is less than or equal to the dimension of the whole spaceClose match
Equivalence theorems for injective transformationsClose match
Nested finite-dimensional subspaces of equal dimension are equalClose match
Definition of automorphism of a vector spaceClose match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
A set of vectors containing more elements than the dimension of the space must be linearly dependentRelated
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Related
A vector can be written uniquely as a linear combination of vectors from independent subspaces.Related

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