Math 311h - Fall 2003
Homework
Assignment 1
- Read sections 1.1-1.6.
- Hand in these problems:
- §1.2 - 35-37
- §1.3 - 3, 18
- §1.4 - 8, 9, 30
- §1.6 - 9, 15, 21
- Due Friday, 12 September.
Assignment 2
- Read sections 2.1-2.5.
- Hand in these problems:
- Show that elementary interchange Ri <-> Rj
can be achieved by operations involving elementary multiplicaion and
elementary modification. (Hint: it can be done in four operations.)
- §2.1A - 5, 16, 17
- §2.2C - 9, 13, 19, 24, 28, 31
- §2.2D - 2, 4, 11
- §2.3D - 6, 10, 28, 33
- Due Friday, 19 September.
Assignment 3
- Read sections 3.1, 3.2.
- Hand in these problems:
- Let a, b, and c be vectors in
R3, and let e1,
e2, and e3 be the standard basis
for R3. Show that if D(a,b,c)
is an alternating scalar-valued function of the three vector
variables a, b, and c, and if D is homogeneous
and additive in each vector argument, then
D(a,b,c) = det(a,b,c)
×
D(e1,e2,e3).
In particular, this shows that if
D(e1,e2,e3)=1,
then D(a,b,c) = det(a,b,c).
- Here is an alternative derivation of Cramer's rule. Start with
the matrix equation Ax = b, where
A = [a1 ... an].
(The aj's are column vectors.) Use the "basic matrix
trick" to write b as a linear combination of the
aj's, and then show that
det(a1 ... b ... an) =
xj det(A),
provided b is in the column j position. Dividing by det(A)
gives Cramer's rule.
- §2.4C - 8, 9, 19, 21-24
- §2.5E - 2, 6, 7, 9, 15, 24
- Due Friday, 26 September.
Assignment 4
- Read sections 3.2-3.3.
- Hand in these problems:
- Let Xa(x) = a ×
x. That is Xa(x) is the cross product
of a into x. Show that Xa(x) is
a linear function from R3 to R3
and find its matrix.
- §3.1 - 3, 4, 7, 9, 10, 13, 16, 23, 24, 33
- Due Monday, 6 October.
Assignment 5
- Read sections 3.3-3.5.
- Hand in these problems:
- §3.2 - 7-11, 13, 26, 27
- §3.3 - 6-8, 15-17, 20, 24, 25, 28, 33, 34, 36
- Due Wednesday, 15 October.
Assignment 6
- Read sections 3.6, 3.7.
- Hand in these problems:
- §3.4 - 7-10, 19, 21ab, 23abc
- §3.5B - 3, 8, 10, 13, 14, 17, 20, 27, 35, 36
40, 41
- §3.5C - 1a, 3, 4, 5
- Due Friday, 24 October.
Assignment 7
- Read section 3.7.
- Hand in these problems:
- §3.5C - 11
- §3.6A - 3, 5, 6, 8, 15, 17
- §3.6B - 2, 3, 8, 11, 13, 19, 21
- Due Friday, 31 October.
Assignment 8
- Read section 3.7.
- Hand in these problems:
- §3.6B - 16, 18
- §3.7A - 2-5, 10, 11, 13
- Due Friday, 7 November.
Assignment 9
- Read section 4.1, 4.4, 5.4, 6.1, 6.2.
- Hand in these problems:
- §3.7B - 3, 4, 7, 10-12, 17
- §3.7C - 1, 2, 5
- The table below contains data obtained by measuring the
concentraion of a drug in a person's blood. Find and sketch the
straight line that best fits the data in the discrete least squares
sense.
Log of Concentration
t | 0 | 1 | 2 | 3 | 4 |
ln(C) | -0.1 | -0.4 | -0.8 |
-1.1 | -1.5 |
- Use the orthonormal basis comprising the normalized Legendre
polynomials (Williamson/Trotter, p. 166) to
obtain the (quadratic) polynomial in P2 that gives the best
continuous least square fit for the function f(x) = e2x on
the interval [-1,1]. Sketch both the function and the polynomial that
fits it.
- Due Wednesday, 19 November.
Assignment 10
- Read sections 7.4D, 8.1, 9.1-9.5.
- Know how to work these problems. Do not hand them in.
- §5.4C - 1, 4, 9, 25, 26
- §6.1C - 8, 9
- §6.2A - 1, 2, 8, 13
- §6.2B - 2, 9
- §7.4D - 21, 23
- §8.1B - 8, 20, 22, 31
Assignment 11
- Read sections 9.1-9.5.
- Know how to work these problems. Do not hand them in.
- §9.1 - 3, 4, 9, 10, 17
- §9.4 - 5, 7, 9, 10, 12-14, 24
- §9.5 - 6, 8, 14, 18, 19, 20
Updated: 4 December 2003 (fjn)