3. Notes

1. In the laws, we really mean that the equations are true for all values of $v, w, u \in V$ and all $r, s \in F$.

2. Notice that (a)-(d) are about addition only (an ``additive group''), while (e)-(h) involve multiplication by scalars.

3. The text's definition of a vector space mentions $F$ as if it is part of the vector space: ``A vector space consists of a field $F$ and a set $V$ such that ...''. That's a little old-fashioned. The difference is only technical.

4. Don't get confused between the zero scalar and the zero vector. I'll usually write the 0 vector as 0 or $\vec 0$. In $\mathbb{R}^2$, for example, these are 0 and 0$= (0,0)$.

5. The operation minus as mentioned above is ``unary'': $-v$. However, it is easy to define binary minus by saying $v-w$ is $v + (-w)$.

6. A binary operation is really a function on pairs. Addition is a function $+:F^2 \rightarrow F$, while multiplication by scalars is a function $\cdot : F \times V \rightarrow V$.

7. In the course it will eventually become clear why all the laws of $\mathbb{R}^n$ follow from the basic laws listed.

8. These various laws together really say that for algebra with scalars and vectors you can ``do what comes naturally''. In this course you are free to do algebra on scalars and vectors without explanation--except when we're talking about proving some laws from others!