Find natural operations on vector spaces

Question

How can you work out the natural operations on a vector space? For example: I know for the vector space $$\mathbb{R}^3 := \{(x,y,z) | x,y,z \in \mathbb{R}\} \text{ over } \mathbb{R}\\ + := (x,y,z)+(a,b,c) = (x+a, y+b, z+c)\\\times := a(x,y,z) = (ax,ay,az)\\\vec{0}:= (0,0,0)$$ But how would you show the natural operations on $\mathbb{C} \text{ over } \mathbb{R}$ or $\mathbb{R} \text{ over } \mathbb{Q}$?

Answer 1

For $\mathbf C$ as an $\mathbf R$-vector space, it is obvious, given one of the standard constructions of $\mathbf C$ – it is defined as $\mathbf R^2$ with componentwise addition and scalar multiplication exactly the same way as the example in your post, plus aptoduct defined as $$ (x,y)(x',y')=xx'-yy',xy'+x'y. $$ In this context, $i$ is just a notation for $(0,1)$ and $1$ for $(1,0)$. They make up a basis for $\mathbf C$, so that we have $$(x,y)=x\cdot 1+y\cdot i,\quad\text{ usually simply denoted }\quad x+iy.$$ As to $\mathbf R$ as a $\mathbf Q$-vector space, it is not finite dimensional over $\mathbf Q$, and this depends on the construction used to define $\mathbf R$. One of the simplest is as equivalence classes of Cauchy sequences of rational numbers for the relation $$u_n\sim v_n\iff \lim_{n\to\infty}(u_n-v_n)=0$$ There are natural termwise operations on the set of sequences of rational numbers, and it can be shown that these operations are compatible with the above equivalence relations, so that we can define natural operations on the quotient space.