5 Linear algebra V

1. New: 9.1.4 Old: 8.1.4. Refer to your previous work on the matrix. For the given matrix solve the system $\dot{\vec x}=A\vec x$ using the same method of diagonalization as used in class. Accurately draw a comprehensive set of typical solution curves in the $x_1$, $x_2$ plane.

2. New: 9.3.6 Old: 8.3.6. Orthogonal means orthonormal. Determine the inverse of the orthogonal matrix using the class procedure only.

3. 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.

4. 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.

5. 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. Write out explicitly the relations that compute the old coordinates from the new ones and vice/versa.