Linear Algebra

Chapter ThreeMaps Between SpacesIIsomorphismsIn the examples following the definition of a vector space we expressed the ideathat some spaces are “the same” as others. For instance, the space of two-tallcolumn vectors and the space of two-wide row vectors are not equal becausetheir elements — column vectors and row vectors — are not equal, but we havethe idea that these spaces differ only in how their elements appear. We will nowmake this intuition 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.That definition wasn’t an end point, instead it led to new insights such as theidea of a basis. Here too, after producing a definition and supporting it, we willget two surprises (pleasant ones). First, we will find that the definition appliesto some unforeseen, and interesting, cases. Second, the study of the definitionwill lead to new ideas. In this way, our investigation will build momentum.I.1 Definition and ExamplesWe start with two examples that suggest the right definition.1.1 Example The space of two-wide row vectors and the space of two-tall columnvectors are “the same” in that if we associate the vectors that have the samecomponents, e.g.,( )1(1 2) ←→2