Please complete the following argument.
We suppose first that we have a combination of the form
$rv + su + tw = 0$
(1)
for some constants $r$, $s$ and $t$. To show that {$v, u, w$} is a linearly independent set, we want to show that
we must have $r = s = t = 0$
To show that $r = 0$, we take the dot product of both sides of (1) with the vector $v$. This gives
$v \cdot (rv + su + tw) = v \cdot 0$
or
$rv \cdot v + sv \cdot u + tv \cdot w = 0$.
(2)
Now, looking at the vectors in $S$ above, we see that
$v \cdot v =
$v \cdot u = 0$
and
$v \cdot w = 0$
Substituting these values into equation (2) gives
$0 = 0$,
