\[\mathbb{R}^2\]
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 \[y=x\]
.