Section |
Questions |
|
1.2 |
6,
7, 8, 9 |
|
1.3 |
2,
5, 7, 9, 11, 15 |
|
1.4 |
1,
2, 5, 7, 9, 10, 11 |
|
1.5 |
1,
3, 4, 7, 9, 11, 12, 13, 22, 23, 24 |
|
|
|
|
2.2 |
1,
2, 3, 4, 5, 6, 7, 8, 16 |
|
|
|
|
3.1 |
1,
2, 3, 4, 5, 6 |
|
3.2 |
1,
2, 3, 4, 5, 7, 8 |
|
3.3 |
3,
4, 6, 7, 8, 11, 12, 13, 15, 16, 17, 18 |
|
3.4 |
1,
2, 3, 4, 5, 11, 14, 21, 22, 23, 24 |
|
|
|
|
4.1 |
1,
2, 4, 6, 16, 17 |
|
4.4 |
1,
2, 3, 4, 5, 7, 8, 9, 13, 14, 15, 16, 17, 19, 23 |
|
4.5 |
2,
3, 4, 5, 6, 8, 9, 10 |
|
|
|
Extra
Problems.
- If A is a
positive definite matrix, is A-1 also
positive definite? Hint: solve
problem 1.3.10 first.
- Show that the sum of
positive definite matrices is positive definite. Is the difference of positive definite matrices positive
definite? What about the product
of positive definite matrices?
- Let l be an eigenvalue for a
positive definite matrix, corresponding to a real and non-zero
eigenvector. Show that l > 0.
- Consider the following
Matrix:
Show that for any non-zero vector x, xTAx
> 0, and thus, A is positive definite.
Hint. Let f(x1,
x2, x3) = xTAx. Then write it as a sum of squares. Use the LU Decomposition, if you wish, to help you get the expression
to look like a sum of squares.
Maple Assignments.
- (Due 23 Feb) Calculate
the equation of the “best fit line” that approximates the following data:
(0,0), (3,8), (4,8), (6,10), (8,12), (10,15), and (12,20). Then compute the error, that is, compute
the vector Ax – b. Also,
calculate ||Ax – b||2.