Linear Algebra

Chapter ThreeMaps Between SpacesIIsomorphismsIn the examples following the definition of a vector space we developed theintuition that some spaces are “the same” as others. For instance, the spaceof two-tall column vectors and the space of two-wide row vectors are not equalbecause their elements — column vectors and row vectors — are not equal, butwe have the idea that these spaces differ only in how their elements appear. Wewill 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 formaldefinition, and then we will produce some results backing our contention thatthe definition captures the idea. We’ve seen this happen already, for instance, inthe first section of the Vector Space chapter. There, the study of linear systemsled us to consider collections closed under linear combinations. We defined sucha 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 insightssuch as the idea of a basis. Here too, after producing a definition, and supportingit, we will get two surprises (pleasant ones). First, we will find that the definitionapplies to some unforeseen, and interesting, cases. Second, the study of thedefinition will lead to new ideas. In this way, our investigation will build amomentum.I.1 Definition and ExamplesWe start with two examples that suggest the right definition.1.1 Example Consider the example mentioned above, the space of two-widerow vectors and the space of two-tall column vectors. They are “the same” inthat if we associate the vectors that have the same components, e.g.,(( )1 1 2 ←→2)153