HW #11, due 3:50pm Friday, May 4

6.7 Inner product spaces

22 (Compute an inner product.)

24 (Compute the norm of a function in an inner product space.)

28 (Use Gram-Schmidt to produce an orthogonal basis for a function space.) Use the mth309.function objects demonstrated in class. Make a plot of your orthogonal basis functions. I am not asking you to obtain expressions in terms of the original given basis.

7.1 Symmetric matrices and quadratic forms

10. (Test if matrix is orthogonal.)

22. (Orthogonally diagonalize a symmetric matrix.)

26. (True or false. And justify.)

40. (Orthogonally diagonalize a symmetric matrix: A = Matrix([[8,2,2,-6,9],[2,8,2,-6,9],[2,2,8,-6,9],[-6,-6,-6,24,9],[9,9,9,9,-21]]) ) The eigenvalues are -30, 30, 15, 6 (with multiplicity 2). Use mth309.nullbasis() to obtain an orthogonal basis for each eigenspace. Caution: nullbasis() does not necessarily return an orthogonal basis for the nullspace.

7.2 Quadratic forms

4. (Matrix of quadratic form.)

16. (Classify quadratic form and obtain diagonal form.)

22. (True or false. And justify.)