Linear Algebra, Isomorphisms

In the examples following the definition of a vector space
we developed the intuition that some spaces are "the same" as others.
For instance, the space of two-tall column vectors and
the space of two-wide row vectors
are not equal because their
elements—column vectors and row vectors—are
not equal, but we have
the idea that these spaces differ only in how their elements appear.
We will now make this idea precise.

This section illustrates a common aspect of a mathematical investigation.
With the help of some examples, we've gotten an idea.
We will next give a formal definition, and then
we will produce some results backing our contention
that the definition captures the idea.
We've seen this happen already,
for instance, in the first section of the Vector Space chapter.
There, the study of linear systems
led us to consider collections closed under linear combinations.
We defined such a collection as a vector space, and
we followed it with some supporting results.

Of course, that definition wasn't an end point,
instead it led to new insights such as the idea of a basis.
Here too, after producing a definition, and supporting it,
we will get two surprises (pleasant ones).
First, we will find that
the definition applies to some unforeseen, and interesting, cases.
Second, the study of the definition will lead to new ideas.
In this way, our investigation will build a momentum.

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