Linear transformations

The transformation matrix and its determinant

Change the vectors or change the blue square by dragging the red points!

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

Drag the red points to transform the curve!

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

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