Linear Algebra, 2020a

Chapter Three
Maps Between Spaces
I
Isomorphisms
In the examples following the definition of a vector space we expressed 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 feel that these spaces differ only in how their elements appear. We will now
make this precise.
This section illustrates a common phase 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.
That wasn’t an end point, instead it led to new insights such as the idea of a
basis. Here also, 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 momentum.
I.1 Definition and Examples
We start with two examples that suggest the right definition.