1
$\begingroup$

If $V$ is a vector space (not the zero vector space) over $\mathbb{R}$, and if $F$ is a finite field. how could I show that it is not possible to define a new scalar multiplication of $F$ on $V$, in a way that $V$ with this scalar multiplication and the usual addition becomes a vector space over $F$?

Thanks in advance.

$\endgroup$
  • $\begingroup$ If $V$ is the zero vector space, it is possible. $\endgroup$ – user14972 Oct 4 '17 at 23:01
  • $\begingroup$ Use the restrictive facts about cardinalities of vector spaces as constrained by the size of the base field. $\endgroup$ – Randall Oct 4 '17 at 23:02
  • $\begingroup$ But what if V is not the zero vector? $\endgroup$ – john fowles Oct 4 '17 at 23:04
2
$\begingroup$

Let the characteristic of $F$ be $p$. Then:

$$\vec 0 = 0 \cdot \vec v = \underbrace{(1+1+1+\cdots+1)}_{p\text{ terms}} \vec v \ne \underbrace{\vec v + \vec v + \vec v + \cdots + \vec v}_{p\text{ terms}} = p\vec v$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.