Answer
The image of the line through the points $\left( {u,v} \right) = \left( {1,1} \right)$ and $\left( {u,v} \right) = \left( {1, - 1} \right)$ is the line through the points $\left( {3,8} \right)$ and $\left( {1,2} \right)$ in the $xy$-plane.
The equation of the line is $y=3x-1$.
Work Step by Step
We have the mapping $G\left( {u,v} \right) = \left( {2u + v,5u + 3v} \right)$.
The image of the point $\left( {u,v} \right) = \left( {1,1} \right)$ is $G\left( {1,1} \right) = \left( {3,8} \right)$.
The image of the point $\left( {u,v} \right) = \left( {1, - 1} \right)$ is $G\left( {1, - 1} \right) = \left( {1,2} \right)$.
Since $G$ is a linear map, the image of a line in $uv$-plane is a line in the $xy$-plane. So, the image of the line through the points $\left( {u,v} \right) = \left( {1,1} \right)$ and $\left( {u,v} \right) = \left( {1, - 1} \right)$ is the line through the points $\left( {3,8} \right)$ and $\left( {1,2} \right)$ in the $xy$-plane.
Next, we find the line equation in slope-intercept form in the $xy$-plane.
The slope is $\frac{{2 - 8}}{{1 - 3}} = 3$. So, the equation of the line:
$y-8=3 (x-3)$
$y=3x-1$
