Summary of lectures and homework assignments
Here is what we have done so far:
- 8/30: We began our discussion of linear algebra, covering much of
sections 1.2-1.4 of Cullen. HW is problems 1, 3, 6, 8, 15 of section
1.3 of Cullen. No due date yet but do it now to keep up.
- 9/1: Cullen 1.4, 1.5. We talked about inverses and
transposes of matrices and the algebraic rules they satisfy. Homework
due Wednesday 9/6 is as follows. For those without a text, here is a copy of the homework problems in
text form.
- 1.3: 1, 3, 6, 8, 15
- 1.4: 2, 4
- 1.5: 1, 2, 5, 7
- 9/6: Cullen 1.5, 1.6, 1.7. We started by talking about
complex numbers, then defined the conjugate transpose which Cullen
calls the tranjugate (not at all standard terminology). We then
spent some time on partitioned matrices. The bottom line here is
that as long as you partition two matrices compatibly, their product is
obtained by pretending the submatrices are scalars. (Just
remember to preserve the order of multiplication.) We did some examples
of calculations using partitioned matrices. Finally we went
through a list of special types of matrices: upper and lower
triangular, diagonal, symmetric, skew symmetric, Hermitian and skew
Hermitian. First part of HW due 9/13:
- 9/8: Cullen 1.8: We described how to solve AX=K. Start with
the partitioned matrix [A K] and do row operations to reach [B H] where
B is in a nice form which allows you to read off the solution (called
the reduced row echelon form). In general there will either be no
solutions, one solution, or infinitely many solutions of the form Xp+Xh
where Xp is one solution and Xh is a solution of
the homogeneous equation AX=0. More HW due 9/13 is:
- 9/11: Cullen 1.9: I made some general remarks about
homework. First, that examples can disprove something but (unless
there are only a finite number of possibilities) examples cannot prove
something. They are useful though for guiding you to an actual
proof by giving you insight into a problem. We also covered some
homework problems. Finally I asserted, but did not justify, that
a row operation corresponds to multiplying on the left by a nonsingular
matrix, in fact the matrix obtained by performing the row operation on
the identity. In fact, although Cullen does not say this
directly, you always have (AB)(ROP) = A(ROP)B.
- 9/13: We spent some time going over problems 1.7 1,4,7 and 1.9
4. I agreed that 1.7 7 was a bit too much and will ask the grader
to grade it as either attempted or not. I will also instruct the
grader to be generous as for as requiring gory details in 1.7 1.
We talked about a number of conditions equivalent to a matrix being
nonsingular. In particular, if A is square and you find B so BA=I
then A is nonsingular and B is the inverse of A. Likewise, if you
find a C so AC=I then C is the inverse of A. Also, A is
nonsingular if and only if AX=0 only has the trivial solution X=0. Next
we did a short computer demo using Matlab. Soon I will give more
details so you can do this yourself. There will also be a project
due probably 9/27 using Matlab. Finally we started talking aboiut
vector spaces, giving the general framework and the examples Fnxm
and Fcn(S,F) given in Cullen, 2.1.
- 9/15: We talked about the determinent. While Cullen devotes
chapter 3 to determinents, I am taking a more abreviated
approach. I gave a handout listing
some properties of determinents and told you how to calculate them
using row operations. You should know how to calculate
determinents of 2x2 and 3x3 matrices and know their properties. I
also handed out your first matlab assignment
which is due 9/27 but you should do it way before then because there
are often problems which are insurmountable if the assignment is left
til the last minute.
- 9/18: We started talking about abstract vector spaces, Cullen
2.1. HW due 9/20:
- Cullen 3.1: 1
- Cullen 3.2: 8 (you may assume the results in problems 3 and 4) Grader's solutions a, b,
c, d (I
apologize for the low quality and large size).
- Colley 1.1: 10, 12, 14, 16 (no need to turn these 4
in, but do them and ask questions if there are problems).
- 9/20: We spent some time on a homework problem. We started
talking about subspaces and the span of a set of vectors, Cullen 2.1
and 2.2. HW due 9/27:
- Cullen 2.1: 2, 5, 8
- Cullen 2.2: 1, 2, 3
- 9/22: We talked more about the span and then introduced linear
dependence and independence. Cullen 2.2 and 2.3.
- 9/25: We talked more about linear independence and bases, Cullen
2.3.
- 9/27: More about bases, we showed that any two bases have the
same number of elements. We gave a method of enlarging any
linearly independent set to a basis of a finite dimensional vector
space, and a way of paring down any spanning set to a basis. We also
defined the rank of a matrix and showed that the rank plus the nullity
equals the number of columns. Cullen 2.3, 2.4
- 9/29: Given a basis for V we showed how to put coordinates in Fkx1
on V. This respects sum and scalar multiplication so it is a
useful way to take an abstract vector space and make it concrete so you
can more easily do calculations on it. Cullen 2.5. I handed
back matlab #1 and post answers to matlab #1.
- 10/2: Linear transformations. We gave some examples and
started talking about one to one and onto. Do the following homework,
don't turn it in but do ask questions on it.
- Cullen 2.3: 4, 5, 8, 10
- Cullen 2.4: 1
- Cullen 2.5: 2
- Cullen 4.1: 1, 2, 9
- 10/4: Mostly we did some problems. I also talked
about the dot product and introduced the cross product. Colley,
1.3, 1.4, and 1.6. Homework due 10/11 and grader's solutions:
- Colley 1.3: 4, 6, 8, 12, 14, 26, 28
- Colley 1.4: 6, 10, 26
- Colley 1.6: 11, 12
- 10/6 Exam #1 Material covered: Cullen 1.1-1.9, 2.1-2.5,
3.1, 4.1 and the handout on determinents. The only field used
will be the real numbers. You need not memorize the definition of
a vector space, but you should know the examples given, Rkxm,
R[x], and Fcn(S,R).
- 10/9: Colley 1.4, 1.2, 1.5. More on dot and cross products.
Equations of lines and planes. More HW due 10/11.
- Colley 1.2: 14, 18, 22
- Colley 1.5: 6, 10
- 10/11: Colley 2.1, 2.2 We did some homework problems and talked
about graphing, limits, and continuity of functions of several
variables. Matlab was used to generate graphs and level sets of
some real valued functions of 2 variables. HW due 10/18 and grader's solutions:
- 2.1: 4, 8
- 2.2: 2, 6, 14, 34, 36, 40, 42
- 10/13: More limits and continuity, Colley 2.2. We showed
how to often show a limit of f does not exist at v by looking at the
limit of f along all lines through v. If the limit along some
line does not exist, or if two lines have different limits then the
limit of f at v does not exist. But if all limits exist and are
equal, that is not quite enough to show the limit of f exists. I
stated that any way of combining continuous functions using composition
and the algebraic rules we have studied gives rise to a continuous
function, as long as you adjust the domain appropriately. Finally
we proved that marix multiplication is continuous, an essential
ingredient of which was the inequality ||Ax|| <= K ||x|| where K is
the square root of the sum of the squares of the entries of A.
- 10/16: We talked about the boundary of a subset of Rn
and defined open and closed subsets of Rn. We then
defined hat it means for a function to be differentiable, namely the
definition given on page 120 of Colley, f is differentiable at a if
there is a matrix L so that f(a+h) = f(a) + Lh + e(h) where limh->0
||e(h)||/||h|| = 0. The point is, if h is close enough to 0 then
f(a) + Lh is a very good approximation to f(a+h). We described
how to calculate the entries of the matrix L, entij(L) = the
partial derivative of the i-th coordinate of f with respect to the j-th
variable.
- 10/18: 2.3, 2.4, 2.5, 2.6. We looked at the chain rule,
directional derivaties, gradient, higher order derivatives, and did
some examples. Note we are skipping the material in 2.4 on Newton's
method. HW due 10/25, problems labelled optional are not to be turned
in, but are odd numbered so you can check your answers with the back of
the book. grader's solutions.
- 2.3: 14, 24, 30 optional: 15, 21, 29
- 2.4: 12 optional: 13
- 2.5: 2, 4, 11, 20 optional: 21
- 2.6: 3, 12, 18 optional: 19
- 2.8: optional 5
- 10/20: 2.6 We talked some more about the chain rule. Then
we talked about implicit functions. Matlab #2
is assigned and is due Monday 10/30. The matlab assignment is
missing instructions for saving the graphics. Here is a technique
which works on the computer lab next door to our classroom. Bring
along a floppy and put it in the computer to save your work. Give
matlab the command cd A:
to tell matlab to do all saves to the floppy. Whenever you want
to save graphic output use the command print
-djpeg Fig1 (or Fig2 for the second figure etc.) which
saves the most recent graphics output onto a file Fig1.jpg on the
floppy. You can then paste this figure into a word
processor document in the appropriate place. Later you can print
out the document on your home computer. Type help print for a list of other
graphics formats you may save in if you don't want jpeg format. If you
have a print account, you can also print graphics at a computer lab by
selecting print in a menu. Maybe people don't use floppies any
more, I know it has been a long time since I have. You should be
able to email the figure to yourself, perhaps first print it out as a
pdf file and attach it to a message. We'll talk about this on
10/27.
- 10/23 5.1, 5.2: We began talking about integration. We
talked about an integral being an infinite sum (or more precisely a
limit of finite sums with more and more terms). We also showed
how to calculate the integral of a function of two variables on a set
of the form a<=x<=b, g(x)<=y<=h(x) for continuous functions
g and h.
- 10/25: 5.3,5.4: We gave some examples switching the order of
integration. We introduced triple integrals. HW due 11/1, starred
problems to be turned in. Do the unstarred ones also, we can
discuss them in class.
- 5.2: 2,4,*6,8,12,14,22,26a optional 3,5,7,9,11
- 5.3: 2,4,*6,8,*10,18, optional 3,5,7,9,11,13,15
- 5.4: 4,6,*12,14,18, optional 7,11,13
- 5.5: 8,*10,12,14,16,*30, optional 11,13,17
- 10/27: 5.5 Change of variable. We learned to translate an
integral to different coordinates. To do this you need to
multiply the integrand by the absolute value of the determinent of the
derivative matrix of the coordinate change (also known as the absolute
value of the Jacobian). Useful examples are polar and cylindrical
coordinates where you multiply by r, and spherical coordinates where
you multiply by rho squared sin phi.
- 10/30: 5.6 We talked about some applications of integrals and did
a bunch of examples.
- 11/1: more examples of setting up integrals
- 11/3 Exam #2 on Colley 1.1-1.6, 2.1-2.6, 5.1-5.6
- 11/6: 3.1, 3.2 Motion in space, velocity, speed, acceleration,
tangential and normal components of acceleration.
- 11/8: More 3.1 and 3.2. There was a quiz at the end of the
period on
integration. Students who got over 80 on exam #2 are exempt from
taking this quiz although they may take the quiz if they wish and
decide whether they want it to count or not (this would only be helpful
if some of your homework or matlab scores were low. Some of your
lowest homeworks will be dropped, but no matlabs). Students who
got less than 80 on exam 2 will have the quiz count as part of the
homework/quiz/matlab score. Some practice problems for the quiz
are 5.8: 1, 3, 5, 6, 9. 10, 11, 19a. The quiz
and its solution are here. Homework due 11/15:
- 3.1: 8, 9, 15, *16
- 3.2: *2, 3, 13, *14, 23, 24, 25, *26
- 3.3: 17, *24
- 3.4: 1,2,3,6,7, *10, 18, 23
- 11/10: We finished up 3.2, defining B and noting that T and N are
not always defined. We started on 3.3, giving examples of vector
fields.
- 11/13: 3.3, 3.4. We talked about div,
grad, and curl.
- 11/15: 6.1, 6.2. We talked about line integrals and
Green's theorem and had a quiz.
Homework due 11/22 (just turn in the starred problems):
- 6.1: *2,3,*6,7,8,11,14,19,22
- 6.2: *2, 4, 6, 12, 17, 19, 20
- 6.3: 2, 3, 4, *8, 12, 14, 18, *20
- 7.1: 4, *20
- 11/17: 6.2, 6.3 We talked about the divergence form of
Green's theorem and the fundamental theorem of line integrals which
makes it easy to calculate work intgerals for a conservative vector
field.
- 11/20: 7.1, 7.2 We talked about scalar and vector surface
integrals.
- 11/22: We did some problems and had a quiz. Homework for 11/29 (do and ask questions
but don't turn in).
- 7.1: 1, 24
- 7.2: 1,2,3,4,710,14,20
- 11/27: We went over the quiz and did a problem on surface
integrals. I mentioned that if curl F = 0, it still might not be
true that F is conservative. However if the domain of F is of a
nice form, then curl F=0 implies that F is conservative. The
condition on the domain of F is that it be simply connected, explained
in Colley. In my notes on div, grad,
and curl I show this in the case where the domain is star shaped
(see the last paragraph on page 1 and the first on page 2) and also
give an example on page 2 of a vector field whose curl is 0 but is not
conservative.
- 11/29: we went over problems, had a quiz,
and started talking about Stokes' theorem (7.3). HW for 12/6 (do
and ask questions
but don't turn in).
- 12/1: Exam #3 on 3.1-3.4, 6.1-6.3, 7.1-7.2. You can bring a
3x5 card with anything on it, front and back.
- 12/4: 7.3 We talked about Gauss' theorem.
- 12/6: We did some problems and had a quiz.
- 12/8: We talked about forms (Chapter
8). This is
optional.
- 12/11: We reviewed linear algebra.
- 12/13: We'll have a review 10-12