A linear transformation between two vector spaces $V$ and $W$ is a map $T:V\rightarrow W$ such that the following hold:
The main example of a linear transformation is given by matrix multiplication. Given an $n\times m$ matrix $A$, define $T(\mathbf v)=A\mathbf v$, where $\mathbf v$ is written as a column vector (with $m$ coordinates). For example, consider \[ A=\left( \begin{array}{cc} 0 & 1 \\ -2 & 2 \\ 1 & 0\end{array} \right) \] then $T$ is a linear transformation from $\mathbb R^2$ to $\mathbb R^3$, defined by $$T(x,y)=(y,-2x+2y,x).$$
The following simulations show the effects of linear transformations, defined by square matrices $A$, on geometric objects.
The determinant of a square matrix is a number that can be related to the area or volume of a region. In particular, the determinant of a matrix reflects how the linear transformation associated with the matrix can scale, rotate, shear or reflect objects (in this case a square of side 2 centred at the origin).
Things to try:
The determinant of a square matrix is a number that can be related to the area or volume of a region. In particular, the determinant of a matrix reflects how the linear transformation associated with the matrix can scale, rotate, shear or reflect objects (in this case a square of side 2 centred at the origin).
Things to try:
The following file contains activities and problems associated with the simulations.
The University of Queensland