$V$ is a vector space. $u,v,w \in V$ and $u+2v+3w=0$. Show that $\mathrm{Span}(u,v)=\mathrm{Span}(v,w)$ or not?
I really struggled with this question. Can anybody help me?
$V$ is a vector space. $u,v,w \in V$ and $u+2v+3w=0$. Show that $\mathrm{Span}(u,v)=\mathrm{Span}(v,w)$ or not?
I really struggled with this question. Can anybody help me?
Hints:
Assuming that char$\,F\neq3\;,\;\;F:=$ the field over which the linear space is defined , check that
$$w=-\frac13u-\frac23v\;\;,\;\;\text{and also}\;\;u=-2v-3w$$
If char$\,F=3\;$ we have a counter example with $\;u=v=0\;,\;\;w\neq0\;$: