vector spaces are isomorphic iff their bases are equipollent

Theorem 1.

Vector spacesMathworldPlanetmath V and W are isomorphic iff their bases are equipollentMathworldPlanetmath (have the same cardinality).


() Let ϕ:VW be a linear isomorphism. Let A and B be bases for V and W respectively. The set


is a basis for W. If


with aiA. Then


since ϕ is linear. Furthermore, since ϕ is one-to-one, we have


hence ri=0 for i=1,,n, since A is linearly independentMathworldPlanetmath. This shows that ϕ(A) is linearly independent. Next, pick any wW, then there is vV such that ϕ(v)=w since ϕ is onto. Since A spans V, we can write


so that


This shows that ϕ(A) spans W. As a result, ϕ(A) is a basis for W. A and ϕ(A) are equipollent because ϕ is one-to-one. But since B is also a basis for W, ϕ(A) and B are equipollent. Therefore


() Conversely, suppose A is a basis for V, B is a basis for W, and |A|=|B|. Let f be a bijection from A to B. We extend the domain of f to all of A, and call this extensionPlanetmathPlanetmath ϕ, as follows: ϕ(a)=f(a) for any aA. For vV, write


with aiA, set


ϕ is a well-defined function since the expression of v as a linear combinationMathworldPlanetmath of elements of A is unique. It is a routine verification to check that ϕ is indeed a linear transformation. To see that ϕ is one-to-one, let ϕ(v)=0. But this means that v=0, again by the uniqueness of expression of 0 as a linear combination of elements of A. If wW, write it as a linear combination of elements of B:


Each biB is the image of some aA via f. For simplicity, let f(ai)=bi. Then


which shows that ϕ is onto. Hence ϕ is a linear isomorphism between V and W. ∎

Title vector spaces are isomorphic iff their bases are equipollent
Canonical name VectorSpacesAreIsomorphicIffTheirBasesAreEquipollent
Date of creation 2013-03-22 18:06:55
Last modified on 2013-03-22 18:06:55
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Result
Classification msc 13C05
Classification msc 15A03
Classification msc 16D40