Chapter 5 continued

1. Execute
   D:=Matrix([[c,0],[0,c]]);
   Part A: Execute the Eigenvectors command.
   Part B: How does this transformation act on R²?
   Part C: Use Part B to explain your output in Part A.

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2. Execute
   Sh:=Matrix([[1,k],[0,1]]);
   Part A: Execute the Eigenvectors command.
   Part B: How does Sh act as a transformation on R² for positive k values?
   Part C: Use Part B to explain your output in Part A.
   Part D: Is A diagonalizable? Why or why not?
   Part E: If k=0, Maple's response to Part A and your response to Part D are both incorrect - what should the responses be?

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3. Execute
   R:=Matrix([[cos(2*theta), sin(2*theta)],[sin(2*theta), -cos(2*theta)]]);
   Notice the difference between this matrix and a rotation matrix.
   Part A: Apply the Eigenvectors(R); and Eigenvalues(R); commands.
   Part B: When theta=0, what geometric transformation is R? What are the eigenvectors for each eigenvalue in this case?
   Part C: When theta=Pi/4, what geometric transformation is R? What are the eigenvectors for each eigenvalue in this case?
   Part D: What is the relationship between theta and the geometric transformation R(theta)?
   Part E: What are the eigenvectors for each eigenvalue for a general theta (hint: the reasoning and trigonometry is similar to what we used for the projection matrix): resolve with the Maple Eigenvectors output.
   Part F: Is R(theta) diagonalizable?