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

Add comment

Security code