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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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- Thread starter Markjdb
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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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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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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.

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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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?Markjdb said:It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?

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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I might be wrong, but it seems to me that in this vector space, any two subspaces without finite bases should be isomorphic

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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.

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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.LukeD said:

I might be wrong, but it seems to me that in this vector space, any two subspaces without finite bases should be isomorphic

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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.

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.

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.

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.

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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