Span

In the section on subspaces, we used set builder notation to define possible candidates for subspaces. There is a better way to do this, however, using something called the span. The span uses a set of distinct, indexed vectors to generate a vector space. How does it do this? It uses every possible linear combination of the set of vectors. As you hopefully see by now, linear combinations are at the heart of linear algebra.

So, let's start with a set, S, of vectors with just one vector, like the following:

Let's look at our one vector graphically:

Figure 4.6 – Graph of the one vector in S

Okay, what would be its span? Or, in other words, what are all the vectors that are linear combinations of this one vector? Well, we can't add because all we have is one vector. So all we can do is scale this vector. If we scale it for all real numbers, it becomes a ….. line! We would say that S spans the space, or span(S) generates...