## Transformation Sending a Line to a Point - or Line

If the determinant of a matrix is zero, the image of a line after transformation by this matrix may or may not be a line in general. It will generally be a point.
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.

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