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(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.

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    The first part was answered [here](http://math.stackexchange.com/questions/2145195/how-to-prove-that-a-beginpmatrix-3-1-endpmatrix-b-beginpmatrix-2/2145201#2145201). Also, you aren't doing it for integer $a$ and $b$. You're doing it for all reals $a$ and $b$. – 2017-02-17
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    For $(b)$ you actually have *two* starting points: "Assume $u,v$ are nonzero vectors such that neither is a scalar multiple of the other" (which FYI means that $u\ne av$ and $v\ne bu$ for any real numbers $a,b$) or "Assume that any two-dimensional vector can be expressed in the form $su+tv$, where $s$ and $t$ are real numbers." – 2017-02-17

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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.