5 10/27

1. New: 9.3.6 Old: 8.3.6. Orthogonal means orthonormal.

2. Find an orthonormal matrix that diagonalizes the matrix
   
   $$\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right)$$
   
   
   Hint: one eigenvalue is 1.

3. New: 9.4.16 Old: 8.4.16 Also accurately draw the conic in the $x_1,x_2$-plane. List the angles of the various axes and asymptotes.

4. New: 9.4.18 Old: 8.4.18 Also accurately draw the conic in the $x_1,x_2$-plane. List the angles of the various axes and asymptotes.

5. Describe the surface $\vec x^T A \vec x = 4$ for the matrix of question 2. What is its shape? How is it oriented?