Elementary Linear Algebra 7th Edition

$S$ spans $R^2$.
Assume the combination of two vectors only of $S$ $$a(-1,4)+b(1,1)=(0,0).$$ We have the system \begin{align*} -a+b&=0\\ 4a+b&=0. \end{align*} Since the determinant of the matrix is given by $$\left| \begin{array} {cc} -1&1\\4&1\end{array} \right|=-5\neq 0$$ then the system has unique solution, that is, $$a=0, \quad b=0.$$ Consequently, the vectors $(-1,4),(1,1)$ are linearly independent and since $R^2$ has the dimension $2$ then they span $R^2$ and hence $S$ spans $R^2$.