Linear transformations
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Find the image of the point \((3,-1)\) under the transformation with the rule: \((x,y) \to (2x + y,x - y)\)
$$
\begin{aligned}
(x, y) & \rightarrow(2 x+y, x-y) \\
\left[\begin{array}{l}
x^{\prime} \\
4^{\prime}
\end{array}\right] &=\left[\begin{array}{cc}
2 & 1 \\
1 & -1
\end{array}\right]\left[\begin{array}{l}
3 \\
-1
\end{array}\right]=\left[\begin{array}{l}
2 \times 3+18-1 \\
3 \times 1+-1 \times-1
\end{array}\right]=\left[\begin{array}{l}
6-1 \\
3+1
\end{array}\right]=\left[\begin{array}{l}
5 \\
4
\end{array}\right] \\
& \therefore \text { image is }(5,4)
\end{aligned}
$$
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