## Transformation Sending a Plane to a Line

If the determinant of a matrix is zero, the image of
$\mathbb{R}^2$
after transformation by this matrix will not be the whole of
$\mathbb{R}^2$
. It will generally be a line.
Consider the transformation of
$\mathbb{R}^2$
represented by the matrix
$M= \left| \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right|$
. Let a general point in
$\mathbb{R}^2$
be
$\begin{pmatrix}a\\b\end{pmatrix}$
then
$\left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) \begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}a+2b\\a+2b\end{pmatrix}$
.
The result of the transformation,
$x=a+2b=y$
and the who;e
$\mathbb{R}^2$
is sent to the line
$y=x$
.