<s>[INST] Show that any 2D vectors can be expressed in the form... 
(a) Show that any 2D vector can be expressed in the form
  $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$
  where $s$ and $t$ are real numbers.
(b) Let $u$ and $v$ be non-zero vectors. Show that any 2D vector can be expressed in the form
  $s u + t v$
  where $s$ and $t$ are real numbers, if and only if of the vectors $u$ and $v$, one vector is not a scalar multiple of the other vector.

No idea what to do to start these problems. Hints appreciated.
 [/INST] (a) Let $\begin{pmatrix} x\\ y\end{pmatrix}$ a 2D vector. Finding $s$ et $t$ such that $$s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix}=\begin{pmatrix} x\\ y\end{pmatrix}$$
is the same thing as solving:
$$\begin{cases}3s+2t=x\\
-s+7t=y\end{cases}$$
where $s$ and $t$ are the unknowns.
(b) Now if you replace $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 7 \end{pmatrix}$ by $u$ and $v$, what are the conditions on the previous system for it to admit a (unique) solution?
</s>