# Linear transformations

## The transformation matrix and its determinant

If the unit vectors \(\mathbf{e_1}\) and \(\mathbf{e_2}\) are mapped onto the vectors \(\mathbf{u}=\binom{a}{b}\) and \(\mathbf{v}=\binom{c}{d}\), then the transformation matrix is given by

\[\mathbf{T}=\left( \begin{array}{} a & c \\ b & d \\ \end{array} \right)\]

The absolute value of the determinant gives the scale factor of the areas. Check out the transformed image of the Mandelbrot set when the determinant is negative!

In order to visualize a linear transformation of a polygon in GeoGebra, do this:

- Make two vectors \(\mathbf{u}\) and \(\mathbf{v}\)
starting at the origin. Do not place the end points on the
`x`- or`y`-axes or they will get stuck there. Rename the end point of \(\mathbf{u}\) to \(A\), and the end point of \(\mathbf{v}\) to \(B\). - Create the transformation matrix \(\mathbf{T}=\left( \begin{array}{} x(A) & x(B) \\ y(A) & y(B) \\ \end{array} \right)\)
- Store the determinant in a variable:
`detT=determinant[T]`

- Make a polygon,
`poly1`

- Transform the polygon by using the command
`ApplyMatrix[T,poly1]`

- Make a variable:
`areas=poly1'/poly1`

. - Right click on
`poly1'`

, choose Properties and then the Advanced tab. Input`detT>0`

in the field ”Blue”, and`detT<=0`

in the field ”Red”. The polygon will become red whenever the determinant is negative. - Observe the variables
`areas`

,`detT`

and the colours of the polygon!

## Linear transformations of curves

The graph of a function \(f(x)\) can be drawn as a curve \(c\) by writing the code:

c=Curve[t, f(t), t, -20, 20]

Each point on the curve has coordinates \((t,f(t))\). Applying a transformation matrix on the corresponding vector will yield the transformed curve. To do it in GeoGebra, create points and vectors as described in the section above. Then enter the curve:

c=Curve[t x(A) + f(t) y(A), t x(B) + f(t) y(B), t, -20, 20]

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License