Linear transformations

Suppose that 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

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

• 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!