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).

Proof.

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

ϕ(A):={ϕ(a)aA}

is a basis for W. If

r1ϕ(a1)++rnϕ(an)=0,

with aiA. Then

ϕ(r1a1++rnan)=0

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

r1a1++rnan=0,

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

v=r1a1++rnan,

so that

w=ϕ(v)=r1ϕ(a1)++rnϕ(an).

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

|A|=|ϕ(A)|=|B|.

() 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

v=r1a1++rnan

with aiA, set

ϕ(v)=r1ϕ(a1)++rnϕ(an).

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:

w=s1b1++smbm.

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

w=s1f(a1)++smf(am)=s1ϕ(a1)++smϕ(am)=ϕ(s1a1++smam),

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