Consider the transformation represented by the matrix
\[M= \left| \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right|\]
. The line in \[\mathbb{R}^2\]
\[y=3x+2\]
may be written \[\begin{pmatrix}x\\3x+2\end{pmatrix}\]
then \[\left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) \begin{pmatrix}x\\3x+2\end{pmatrix}=\begin{pmatrix}7x+4\\14x+8\end{pmatrix}\]
.The result of the transformation is that the
\[y\]
coordinate is twice the \[x\]
coordinate so the equation of the line is \[y=2x\]
.Consider the effect of the transformation on the line
\[x=-2y+1\]
, which may be represented by \[\begin{pmatrix}-2a+1\\a\end{pmatrix}\]
.\[\left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) \begin{pmatrix}-2a+1\\a\end{pmatrix}=\begin{pmatrix}1\\2\end{pmatrix}\]
.In this example a line is sent to a point.