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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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Finite-dimensional vector spaces of equal dimension over the same field are isomorphic

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Isomorphic vector spaces have the same dimension.Close match
IsomorphismClose match
Definition of isomorphic/isomorphism between vector spacesClose 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
Nested finite-dimensional subspaces of equal dimension are equalClose match
Equivalence theorems for injective transformationsClose match
DimensionClose match
Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.Close match
The dimension of a subspace is less than or equal to the dimension of the whole spaceClose match
Axioms of a vector spaceClose match
Definition of automorphism of a vector spaceClose match
The inverse of an isomorphism is an isomorphism.Close match
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.Close match
Definition of dimension of a vector space (or subspace)Related
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Related
Definition of dimension of a vector space (or subspace) being finite or infiniteRelated
A vector can be written uniquely as a linear combination of vectors from independent subspaces.Related
If a vector space has dimension nRelated

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