(a) Show that any two-dimensional 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 two-dimensional 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.
I know that we have to prove that $$3s+2t=a$$ $$-s+7t=b$$ for any integers a and b. But I don't know how to prove it. As for part(b), I do not have a starting point.
For part (a) see that the two vectors $\begin{pmatrix} 3 \\ -1 \end{pmatrix} ,\begin{pmatrix} 2 \\ 7 \end{pmatrix}$ are linearly independent and hence form a basis of the set of all two-dimensional vector.