Answer
See below
Work Step by Step
Given the set of vectors $\{(1,1),(-1,2),(1,4)\}$
Rewrite $v=(1,4)=2(1,1)+(-1,2)$
Assume for every $v=(x,y) \in R^2$ there are constants $c_1$ and $c_2$ such as:
$(x,y)=c_1(1,1)+c_2(-1,2)$
We have the system:
$c_1-c_2=x\\
c_1+2c_2=y$
The determinant of the matrix can be determined as $ \det=\begin{vmatrix}
1 & -1 \\1 & 2
\end{vmatrix}=2-(-1)=3\ne 0$
Hence, the matrix is invertible and the system below has unique solution.
There are constants $c_1$ and $c_2$ such as
$(x,y)=c_1(1,1)+c_2(-1,2)$
Vector $(1,1)$ and $(-1,2)$ spans $R^2$.
Hence, set of vectors also $\{(1,1),(-1,2),(1,4)\}$ spans $R^2$
