How to prove the Span of a set $S$ is equal to $\Bbb{R}^2$.

Question

absolutely baffled. need to prove $\operatorname{Span}\{(2,1), (-1,3), (0,1)\} = \Bbb{R}^2$,

I started by showing that $(2,1)$ can be written as $-2(-1, 3) + 7(0,1)$ but thats it so far

Answer 1

Since $(0,1)$ belongs to the set, you just need to show that $(1,0)$ belongs to the span, which is clear enough: $$ (1,0)=\frac{1}{2}\bigl((2,1)-(0,1)\bigr) $$ More systematically, consider the matrix $$ \begin{bmatrix} 2 & -1 & 0 \\ 1 & 3 & 1 \end{bmatrix} $$ and perform Gaussian elimination to show that the matrix has rank $2$.