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Linear Transformations

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-07)

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(xml+mathml Lecture-07)

Visualization of linear transformations

1. Take a polyhedron (an octahedron, for example) and color each of the faces and display it.
2. Apply the matrix:

   $$\left( \begin{array}{ccc} 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & -1\ \end{array} \right)$$ (07-15)

   
   
   to each of the vertices. Note that the transformation reflects along the z-directions across the x-y plane.
3. Apply the matrix:

   $$\left( \begin{array}{ccc} 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & 5\ \end{array} \right)$$ (07-16)

   
   
   to each of the vertices.
   

4. Apply the matrix:

   $$\left( \begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0\ \sin(\theta) & \cos(\theta) & 0\ 0 & 0 & 1\ \end{array} \right)$$ (07-17)

   
   
   to each of the vertices. Its determinant is unity.