Spans

Sometimes mathematicians are lazy. We want to be rigorous and accurate, but sometimes we really just want to say “Come on! You know what I mean.” So we have a way to do that. Last time I gave an example of an “exotic” subspace of \mathbb C (thought of as a vector space over \mathbb R. It was:

\{a\cdot\pi + a\cdot i\mid a \in \mathbb R\}

But, come on. That’s a lot to write. Can’t I just say “you know, the subspace that has the vector \pi+i? Strictly speaking, no. There is more than one such subspace, but we can do pretty much that. I can say the subspace spanned by \pi +i. You can think of this in two equivalent ways:

  1. Think of it as the smallest subspace containing \pi + i.
  2. Think of it as taking \pi + i and then whatever else you need to make it a vector space.
Another example, the subspace of \mathbb R^3 spanned by (2.5,3,0). What else must we have. We need exactly the points a\cdot (2.5,3,0) for every a. That is,
\mbox{span}((2.5,3,0))=\{(2.5a,3a,0)\mid a\in\mathbb R\}.
We use the notation \mbox{span}(\cdot).
But it gets more complicated when you have more points. What if I want to say “you know what I mean” about the points p=(1,0,0) and q=(0,0,1) in \mathbb R^3? That is, what is \mbox{span}(\{p,q\})?
Well, we need everything that looks like (a,0,0) and everything that looks like (0,0,b). But we need even more. We need every possible combination of these two. That is, to be a subspace, we need to be able to add any two vectors and get another one in the subspace. So it turns out, hopefully not surprisingly, that the subspace we want is
\mbox{span}(\{p,q\})=\{(a,0,b)\mid a,b\in\mathbb R\}.
You can imagine these sorts of things can get very complicated, and so having this notation makes talking about subspaces simpler.
One more thing to whet your appetite for what is to come. In \mathbb R^3, \mbox{span}(\{(0,1,-1),(0,-\pi,\pi)\})=\mbox{span}((0,1,-1)). Make sense?
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