Notes

This is to explain further what I said in class about the laws of vector spaces. Since the kind of reasoning involved is a little advanced, this won't be on the midterms or final, but it's worth looking at to see what the issues are.

The laws used in the definition of a vector space are some of the laws true in $F^n$ for every $n$.

Additional laws not part of the definition include basic ones such as $0 v =$   0 but also miscellaneous ones that could be proved, such as $(r+s)(v+w) = rw + (sv + (sw + rv))$.

Near the beginning of the course I mentioned this fact:

Proposition. The defining laws of vector spaces over R are enough to imply all the laws that hold in $\mathbb{R}^n$ for all $n$, in the sense that any vector space does satisfy all these laws.

(The same is true for any field $F$ in place of $\mathbb{R}$, but let's stick to $\mathbb{R}$.)

At the time it was not clear how a statement like this could be proved. But now it can be. First, a proof that doesn't quite work:

?? ``Proof.'' If $V$ is a vector space over $\mathbb{R}$, then $V \cong \mathbb{R}^n$, so all the laws of $\mathbb{R}^n$ hold in $V$.

The trouble with this attempted proof is that a vector space might not be finite dimensional and so might not be isomorphic to $\mathbb{R}^n$, no matter how $n$ is chosen. (Notice that in the attempted proof, $n$ comes out of nowhere.)

Valid proof. Given any law that holds in $\mathbb{R}^n$ for all $n$, we want to try to verify it in $V$. The law has a certain number of variables, say $m$. We could write the law using variable symbols $v _ 1,\ldots, v _ m$. To show that the law holds in $V$, we need to check that the law is true for all possible ways of putting specific vectors for $v _ 1,\ldots, v _ m$. With specific vectors $v _ 1,\ldots v _ m$, we can take the subspace $W$ that they span. We know that $W$, being the span of a finite list of vectors, is finite-dimensional, say of dimension $n$ (with $n \leq m$). Then $W \cong \mathbb{R}^n$, so the law in question, being true in $\mathbb{R}^n$, is true in $W$ for those specific vectors.

That's the proof. Notice that each time a different law and list of specific vectors is checked, a different $W$ is used.