Theorem
If $x,y\in \mathbf{R}^2$ and one is not a multiple of the other, then every vector in $\mathbf{R}^2$ is a linear combination of $x$ and $y$.
Proof
Indeed, as $x$ and $y$ are not multiples of one another, then $x\neq\vec{0}$ and $y\neq\vec{0}$. We can also see that the first coordinates of the two vectors can’t both be zero. Assume without loss of generality that $x_1\neq0$.
Let $z\in \mathbf{R}^2$ be any vector. If $y_2x_1=y_1x_2$, then $y=\frac{y_1}{x_1}x$, i.e., $y$ is a multiple of $x$, which contradicts the hypothesis. Therefore, we can take $\alpha_2=(z_2x_1-z_1x_2)/(y_2x_1-y_1x_2)$ and $\alpha_1=(z_1-\alpha_2y_1)/x_1$ so that $z=\alpha_1x+\alpha_2y\qquad\square$
