Linear Algebra/Definition and Examples of Isomorphisms

Template:Navigation We start with two examples that suggest the right definition.

(Template:Anchor"Morphism" means map, so "isomorphism" means a map expressing sameness.)

Template:AnchorWe are sometimes interested in an isomorphism of a space with itself, called an automorphism. An identity map is an automorphism. The next two examples show that there are others.

As described in the preamble to this section, we will next produce some results supporting the contention that the definition of isomorphism above captures our intuition of vector spaces being the same.

Of course the definition itself is persuasive: a vector space consists of two components, a set and some structure, and the definition simply requires that the sets correspond and that the structures correspond also. Also persuasive are the examples above. In particular, Example 1.1, which gives an isomorphism between the space of two-wide row vectors and the space of two-tall column vectors, dramatizes our intuition that isomorphic spaces are the same in all relevant respects. Sometimes people say, where $V\cong W$, that "$W$ is just $V$ painted green"—any differences are merely cosmetic.

Further support for the definition, in case it is needed, is provided by the following results that, taken together, suggest that all the things of interest in a vector space correspond under an isomorphism. Since we studied vector spaces to study linear combinations, "of interest" means "pertaining to linear combinations". Not of interest is the way that the vectors are presented typographically (or their color!).

As an example, although the definition of isomorphism doesn't explicitly say that the zero vectors must correspond, it is a consequence of that definition.

The definition of isomorphism requires that sums of two vectors correspond and that so do scalar multiples. We can extend that to say that all linear combinations correspond.

In addition to adding to the intuition that the definition of isomorphism does indeed preserve the things of interest in a vector space, that lemma's second item is an especially handy way of checking that a map preserves structure.

We close with a summary. The material in this section augments the chapter on Vector Spaces. There, after giving the definition of a vector space, we informally looked at what different things can happen. Here, we defined the relation "$\cong$" between vector spaces and we have argued that it is the right way to split the collection of vector spaces into cases because it preserves the features of interest in a vector space—in particular, it preserves linear combinations. That is, we have now said precisely what we mean by "the same", and by "different", and so we have precisely classified the vector spaces.

Exercises

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Footnotes

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