I'm new in the University and I don't know how to solve this:
Suppose $v$ is a non null element of a vector space $V$ on $\mathbb R$. Show that the function is injection:
$\mathbb R\to V $
$t \mapsto tv$
Sorry for any misspelling.
Suppose $tv = sv$, then $(t-s)v = 0$. As $v \neq 0$ (that is, it is not the null element of $V$), we must have $t - s = 0$, so $tv = sv \Rightarrow t = s$. Therefore, the mapping is injective.