Vector space isomorphism

In summary, the conversation discusses finding an example of a vector space V that is isomorphic to a proper subspace W. The example given is V=C[x,y] with a basis of polynomials and a bijection to another set, which shows that V is isomorphic to a proper subspace. It is also mentioned that other examples can be found in infinite dimensional vector spaces, such as the space of infinite sequences of real numbers.
  • #1
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I came across this problem today and haven't been able to figure it out...

Give an example of a vector space V which isomorphic to a proper subspace W, i.e. V != W.

It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?
 
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  • #2
Polynomials over a field

Let V=C[x,y]. A basis for this space is [tex] $ B= \{ x^i y^j \mid i,j=0,1,2,...\} $ [/tex]. It is well known that there is a bijection [tex] $ f: Z_{+} \times Z_{+} \rightarrow Z_{+} $ [/tex]. Therefore, if we let [tex] $ t_{i,j} =x^iy^j \ \forall i, j $ [/tex], then we have a bijective map from B to the set [tex] $ B' = \{T^k \mid k \in Z_{+}\} $ given by $ F(t_{i,j}) = T^{f(i,j)} $ [/tex]. Clearly F linearly extends from the basis to all of V and is an isomorphism onto C[T]. You then may trivially send C[T] to C[x] via the isomorphism T -> x.
 
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  • #3
Markjdb said:
It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?
I assume you've considered infinite dimensional vector spaces; where did you run into difficulty showing that one might be isomorphic to a proper subspace?

Analogy might help -- can you think of any other infinitary structure that is isomorphic to a proper substructure? What about the simplest kind of structure: that of simply being a set?
 
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  • #4
Even easier, consider the vector space of infinite sequences of real numbers (or equivalently, countably infinite tuples)

I might be wrong, but it seems to me that in this vector space, any two subspaces without finite bases should be isomorphic
 
  • #5
any bijection between bases yields and isomorphism between the spaces, so just find a basis and a bijection with a proper subset.

e.g. if the basis is the natural numbers, the usual bijection (x-->x+1)with those > 1 induces the famous "shift operator" on the space of (finite) sequences.
 
  • #6
LukeD said:
Even easier, consider the vector space of infinite sequences of real numbers (or equivalently, countably infinite tuples)

I might be wrong, but it seems to me that in this vector space, any two subspaces without finite bases should be isomorphic
The sequences with only finitely many nonzero terms form a subspace that admits a countably infinite basis, whereas the space of absolutely summable sequences (l_1) has as its dimension the cardinality of the continuum. So these two spaces aren't isomorphic. But on the other hand, all the l_p spaces (for 1<=p<infinity) are isomorphic as vector spaces, and l_p is a proper subspace of l_q when p<q.
 
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1. What is a vector space isomorphism?

A vector space isomorphism is a type of linear transformation between two vector spaces that preserves the structure and properties of the vector spaces. It is a one-to-one mapping that preserves addition and scalar multiplication operations between the two vector spaces.

2. How is vector space isomorphism different from other types of linear transformations?

Unlike other linear transformations, vector space isomorphisms are bijective, meaning they have both a one-to-one and onto mapping between vector spaces. This ensures that no information is lost during the transformation.

3. What are the properties of vector space isomorphisms?

Vector space isomorphisms have several important properties, including preserving linear independence, spanning sets, and dimensionality. They also preserve linear combinations and linear transformations between vector spaces.

4. How can vector space isomorphisms be represented?

Vector space isomorphisms can be represented in several ways, including using matrices, equations, and geometric transformations. The most common representation is using matrices, where the transformation is represented by a square matrix with the same dimensions as the vector spaces.

5. What is the importance of vector space isomorphisms in mathematics and science?

Vector space isomorphisms are essential in many areas of mathematics and science, including linear algebra, geometry, and physics. They allow for the study of vector spaces in a more abstract and general way, making it easier to understand and solve complex problems.

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