<s>[INST] A problem with two dimensional vectors and scalar multiples 
Let $\vec{x}$ and $\vec{y}$ be non-zero vectors. Show that any two-dimensional vector can be expressed in the form
  $s \vec{x} + t \vec{y}$, where $a$ and $b$ are real numbers, if and only if of the vectors $\vec{x}$ and $\vec{y}$, one vector is not a scalar multiple of the other vector.

I am not quite sure how to approach this problem. Must the vectors $\vec{x}$ and $\vec{y}$ parallel for this to be true? I just have a suspicion, but I am actually struggling to find a good place to begin.
 [/INST] Suppose $\vec{x} = k \vec{y}$. Then, $s\vec{x} + t\vec{y} = (s+kt)\vec{x}$. The locus of such points as $s$ and $t$ range over the reals is a line. So, not every two-dimensional vector can be expressed as $$s\vec{x} + t\vec{y}.$$For the other direction, let $\vec{x} = \begin{pmatrix} a \\ b \end{pmatrix}$ and $\vec{y} = \begin{pmatrix} c \\ d \end{pmatrix}$. Let $\begin{pmatrix} w \\ z \end{pmatrix}$ be an arbitrary two-dimensional vector.
We want to find real numbers $s$ and $t$ such that
$$s \begin{pmatrix} a \\ b \end{pmatrix} + t \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} w \\ z \end{pmatrix}.$$This gives us the system of equations
\begin{align*}
as + ct &= w, \\
bs + dt &= z.
\end{align*}Solving this system in $s$ and $t$, we find
$$s = \frac{dw - cz}{ad - bc} \quad \text{and} \quad t = \frac{az - bw}{ad - bc}.$$
Both of these quantities are well-defined as long as $ad \neq bc$. Thus, we can conclude if we show that $ad = bc$ implies that of the two non-zero vectors $\vec{x} = \begin{pmatrix} a \\ b \end{pmatrix}$ and $\vec{y} = \begin{pmatrix} c \\ d \end{pmatrix}$, one vector is a scalar multiple of the other vector.
So, suppose $ad = bc$, If $a \ne 0$, then $d = \frac{c}{a} \cdot b$, so $$\frac{c}{a}  \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} c \\d \end{pmatrix}.$$ The case where $c \ne 0$ is similar.
If $a= c =0$, then both vectors are of the form $\begin{pmatrix} 0 \\ z \end{pmatrix}$, so one must be a multiple of the other.
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