| Date due |
Assignment |
| F 1/13/23 |
Assignment 0 (just reading, but important to
do before the end of Drop/Add)
Read the Class home page,
and Syllabus and course information
handouts.
Read the Homework Rules earlier
on this page.
Go to the Miscellaneous Handouts
page and read the handouts "What is a proof?" and "Mathematical
grammar and correct use of terminology". (Although this course's prerequisites
are supposed to cover most of this material,
most students still
enter MAS 4105 without having had sufficient
feedback on their work to eliminate common mistakes or bad habits.)
I
recommend also reading the handout
"Taking and Using Notes in a College Math Class,"
even though it is aimed at students in Calculus 1-2-3 and Elementary
Differential Equations.
Read these tips on using your
book.
In FIS, read Appendix A (Sets) and Appendix B (Functions).
Even though this material is supposed to have been covered in MHF3202
(except for the terminology and notation for images
and preimages in the first paragraph of Appendix B
that you're not expected to know yet),
you'll need to have it at your fingertips. Most students entering
MAS4105 don't.
Also read the
handout "Sets and Functions" on the Miscellaneous Handouts
page, even though some of it repeats material in the
FIS appendices.
I originally
wrote this for students who hadn't taken MHF3202 (at a time when
MHF3202 wasn't yet a prerequisite for MAS4105), so the level may
initially seem very elementary. But don't be fooled: these
notes include some material that most students entering MAS4105
are, at best, unclear about, especially when it comes
to writing mathematics.
For the portions of my handout
that basically repeat what you saw in FIS Appendices A and B,
it's okay just to skim.
In the book by Hammack that's the first item on the
Miscellaneous Handouts page, read Section 5.3 (Mathematical
Writing), which has some overlap with my "What is a proof?" and
"Mathematical grammar and correct use of terminology". Hammack
has a nice list of 12 important guidelines that you
should
already be following, having completed MHF3202. However,
most students entering MAS4105 violate almost all of these
guidelines. Be actively thinking when you read these
guidelines, and be ready to incorporate them into your
writing.
Expect to be penalized for poor writing otherwise.
I'd like to amplify guideline 9, "Watch out
for 'it'." You should watch out for any
pronoun, although "it" is the one that most commonly causes
trouble. Any time you use a pronoun, make sure that it has
an clear and unambiguous antecedent.
(The antecedent of a pronoun is the noun that the pronoun
stands for.)
|
| T 1/17/23 |
Assignment 1
Read Section 1.1 (which should be review).
1.1/ 1–3, 6, 7. Note:
The book's term "the equation of a line" is
misleading, even if the word "parametric" is included, since
the same line can be described by more than one parametric
equation. In fact, there are infinitely many
parametric equations for the same given line. All of the
above applies to planes as well as lines.
Any time I say something in class like
"check this for homework" (e.g. checking that all the vector-space
properties were satisfied in an example), do that checking before
the next class. An item I assign this way at the
blackboard won't be part of the hand-in homework, unless it
happens to coincide with one of the numbered exercises, so (except
in the latter case) it's not something you have to worry about
writing up neatly.
In Section 1.2, read up through at least Example 3 before the
Wednesday Jan. 11 class,
and read the remainder of this section before the Friday Jan. 13 class.
Remember that whenever the book refers to a general field \(F\), you may
assume \(F={\bf R}\) (in this class) unless I say otherwise. If an
exercise I've assigned refers specifically to the field \({\bf C}\)
(plain "C " in the book) of numbers (e.g. 1.2/ 14) then you
need to use \({\bf C}\) as the problem-setup indicates.
Everything in Section 1.2 works if \({\bf R}\) if replaced by \({\bf
C}\);
no change (other than notational) in any definition or proof is needed.
1.2/1–4, 7, 8, 10, 12–14, 17–21.
In #1, the
instructions should have said that, in this exercise, the
letters \(a\) and \(b\)
represent
scalars, while the letters \(x\) and \(y\) represent vectors.
In #14, remember that
"C" is the book's notation for the set of complex numbers.
As the book mentions, \(C^n\), with the indicated operations, is
a complex vector space (a vector space over the field
\(C\)). However, what you're being asked in #14 is whether \( C^n
\), with the indicated operations, is also a real vector
space.
Note: To show that a given set with given operations
is not a vector space, it suffices to show that one
of the properties (VS 1)–(VS 8) does not hold. So in each
of the exercises in the 13–20 group, if the object you're
asked about turns out not to be a (real) vector space, an
appropriate answer is "No; (VS n) does not hold"
(where n is an appropriate one of the numbers 1, 2, ...,
8), together with an example in which (VS n) fails. Even if
two or more of (VS 1)–(VS 8) fail, you only need to
mention one of them that fails. In your own notes
that you keep for yourself, it's fine if you record your analysis
of each of the properties (VS 1)–(VS 8), but don't hand
that in.
However, to show that something is
a vector space,
you have to show that each of (VS 1)–(VS 8)
holds.
On Tuesday 1/17, hand in only the
following problems:
- 1.1/ 2d, 3b, 6
- 1.2/ 4bdf, 14, 18, 19, 21
In #14, if the answer is no, give a
reason why. If the answer is yes, state why the
vector-space properties that involve scalar multiplication
hold. (In this problem, a full proof checking (VS1)-(VS8) is
not required.)
In #21, the vector space \({\sf Z}\) is
called the direct sum of \({\sf V}\) and \({\sf
W}\), usually written \( {\sf V} \oplus {\sf W}
\). (Sometimes the term "external direct sum" is
used, to distinguish this "\( {\sf V} \oplus {\sf W} \)"
from something we haven't defined yet that's also called a
direct sum.)
|
| T 1/24/23 |
Assignment 2
Read Section 1.3
Exercises 1.3/ 1–10, 13, 15, 19, 22
In #22, assume \({\bf F}_1={\bf F}_2={\bf
R}\), of course.
Do the following, in the order listed.
- Read the first definition—the definition
of \(S_1+S_2\)—near the bottom of p. 22. (The second
definition is correct, but not complete; as you saw in the
previous assigment, there is something else that's also called
direct sum. Both types of direct sum are discussed and compared in
a handout later in this assignment.)
- Exercise 1.3/ 23.
In part (a), if we were to insert a
period after \(V\), we'd have a sentence saying, "Prove that
\(W_1+W_2\) is a subspace of \(V\)." Think of this as
part "pre-(a)" of the problem. Obviously, it's something
you'll prove in the process of doing part (a), but I want the
conclusion of "pre-(a)" to be something that stands out in your
mind, not obscured by the remainder of part (a).
- Read the short handout "Direct Sums" posted on the
Miscellaneous Handouts page.
- Do exercises DS1, DS2, and DS3 in the Direct Sums handout.
- Exercises 1.3/ 24, 25, 26, 28–30. See additional instructions below about
24, 28, and 30.
- #24: In addition to what the book says to do,
figure out how #24 is related to exercise DS1.
- In #28, skew-symmetric is a
synomym for antisymmetric. You're not required to know
what a field of characteristic two is; you may just take
it on faith that \({\bf R}\) isn't such a
field. (But in case your interest was
piqued: every field has a characteristic, which is either
0 or a prime number. A field \({\bf F}\) of characteristic
\(p>0\) has the property that \(px=0\) for all \(x\in {\bf F}\),
where \(px=\underbrace{x+x+\dots +x}_{p\ \
\mbox{times}}\).)
The second part of #28 may
be the first proof you've been asked to do, in this class, in
which you have to come up with an idea that nobody hss
shown you before. As you go further in math, problems with this
feature will become more and more prevalent. There is no general
prescription for coming up with an original idea. However, one
approach that's often helpful (for generating ideas, not
for writing the eventual proof) is to "work backwards". In this
problem, for example, showing that \(W_1+W_2=M_{n\times n}({\bf
R})\) amounts to showing that for every \(n\times n\) matrix
\(B\), there exist an antisymmetric matrix \(A\) and a symmetric
matrix \(S\) such that \(B=A+S\). If the statement we're being
asked to prove is, indeed, true, then by the result we're asked
to prove in #30, there will be a unique such \(A\) and
\(S\), so perhaps we can find an actual formula that
produces \(A\) and \(S\) from \(B\). Since the definition
of symmetric and anti-symmetric matrices involves
transposes (I didn't define
"transpose" in class, or use it to define symmetric or
antisymmetric matrices, but the definition I gave is equivalent
to the one in the book), maybe if we play with the
equation "\(B=A+S\)" and the transpose operation, we can figure
out what \(A\) and \(S\) need to be.
-
In #30, just prove the half of
the "if and only if" that's not exercise DS3.
On Tuesday 1/24, hand in only the
following problems:
- DS1–DS3
- 1.3/ 2dfh, 3, 6, 10, 15, 19, 23
part "pre-(a)" (see above), 24, 28, 30
|
| T 1/31/23 |
Assignment 3
Before the Wed. 1/25/23 class, read from the beginning of
Section 1.4 up through the definition of span on p. 30, feeling
free to skip Example 1. (You are certainly welcome to read
Example 1, but I've never read it and don't plan to!)
Note: in the three procedural steps below "The
procedure just illustrated" on p. 28, the 2nd and 3rd steps should have
been stated more precisely. In the illustrated procedure, each step
takes us from one system of equations to another system with the
same number of equations; after doing step 2 or 3, we don't simply
append the new equation to the old system. The intended meaning of
Step 2 is, "multiplying any equation in the system by a nonzero
constant, and replacing the old equation with the new
one." The intended meaning of Step 3 is, "adding a constant
multiple of any equation in the system, say equation A, to another
equation in the system, say equation B, and replacing
equation B with the new equation."
Of course, these intended meanings are
clear if you read the examples in Section 1.4, but the authors still
should have stated the intended meanings explicitly.
Before the Fri. 1/27/23 class, read the remainder of Section 1.4.
1.4/ 1 (all parts), 3abc, 4abc, 5cdegh, 6, 8–16.
In 5cd, the vector space in consideration is \({\bf R}^3\); in 5e
it's \(P_3({\bf R})\); in 5gh it's \(M_{2\times 2}({\bf R})\).
In 1de:
- Interpret the indicated operation as replacing an
equation by one obtained by the stated operation. In 1e, the
equation being replaced is the one to which a multiple of another
equation was added.
- The intended meaning of "it is permissible" to do the
indicated operation is that that operation never changes the
solution-set of a system of equations. No permission from
your professor (or other human being, alien overlord,
fire-breathing dragon, etc.) is involved.
On Tuesday 1/31, hand in only the
following problems:
- 1.4/ 3b, 4b, 5dh, 6, 10, 14, 16.
I apologize for not getting these posted Friday night, as
thought I'd done! Hopefully, with the hand-in list
being fairly short, and your having already worked
out all the assigned problems (not just the
hand-in ones) on scratch paper, you shouldn't have too
much trouble writing up the hand-in problems neatly, in
my required format, with just 22 hours' notice.
|
| T 2/7/23 |
Assignment 4
Read Section 1.5.
1.5/ 1, 2a–f,
3, 4, 6, 7, 10, 12, 13, 15–17, 20, 21
In Section 1.6, read at least as far as Theorem 1.9 before
Friday's class (Feb. 3). Read the rest of Section 1.6, minus
the subsection on the Lagrange Interpolation Formula (we're
skipping that) and Example 11 (which you may treat as optional
reading), by Monday's class.
On Tuesday 2/7, hand in only the
following problems:
- 1.5/ 2bdf, 3, 6, 10, 13b, 15, 17, 20
Note: the book's exercises from a given section are
always intended to be done using just the material
covered up through that section; you're never allowed
to forward-reference and use results, terminology, etc. from later
sections. In particular, the words "basis" and "dimension",
and any result proved in Section 1.6, should not appear in your
solutions to Section 1.5 exercises.
|
| T 2/14/23 |
Assignment 5
Read Section 1.6.
1.6/ 1–8, 12–18 (note: in #18, \({\sf W}\) is not
finite-dimensional!), 21, 25 (see below), 29, 33, 34.
For several of these problems (for example, #4 and #12), results
proven in class (and in Section 1.6) can (and should) be
used to considerably shorten the work needed.
For students who have taken MAS3114 (Computational Linear
Algebra): See the note in the previous assignment about
forward-referencing. In FIS, matrix methods (row reduction,
etc.) for solving systems of linear equations aren't covered
until Chapter 3. Until we cover those methods in this
course, you're not allowed to use them in any work you hand in.
Part of what we'll prove in this class is the validity of those
methods, which may have been touched in on MAS3114, but that I
can't assume you've seen a proof of. Also, if you make any
mistakes, it's very difficult for a grader to comment on what
you did wrong when we haven't covered a technique yet.
Note about exercise 25.
Another way to word this exercise is: For arbitrary
finite-dimensional vector spaces \(V\) and \(W\), express the
dimension of the external direct sum \(V\oplus_e W\) in terms
of \({\rm dim}(V)\) and \({\rm dim}(W).\)
But note that both this wording the
book's have a deficiency: since we have defined "dimension" only
for finite-dimensional vector spaces, we really shouldn't
even refer to "the dimension of \(V\oplus_e W\)" (the
dimension of \(Z\), in the book's wording) without first knowing
that \(V\oplus_e W\) is finite-dimensional. The second
half of my alternate wording should, more sensibly, have said,
"show that the external direct sum \(V\oplus_e W\) is
finite-dimensional, and express its dimension in terms of \({\rm
dim}(V)\) and \({\rm dim}(W).\)
However, this "sensible" wording of
the exercise has a
practical drawback: it can
lead students to expect that the work they do must,
effectively, have a "part (a)" (showing finite-dimensionality of
the direct sum) and a "part (b)" (giving a formula for
the dimension), when in fact these parts end up being done
simultaneously. To do #25, start with bases of \(V\)
and \(W\), make an educated guess that a certain finite subset
of \(V\oplus_e W\) (with easily computed cardinality) is a basis,
and then show that your basis-candidate is indeed a basis. That
shows, simultaneously, that \(V\oplus_e W\) has a finite basis
(hence is finite-dimensional) and that the cardinality of your
candidate-basis is the dimension of \(V\oplus_e W\).
Do these non-book problems.
On Tuesday 2/14, hand in only the
following problems:
- 1.6/ 3bd, 12, 14, 16, 25, 29,
34a
- non-book problem NB 5.3
|
| T 2/21/23 |
Assignment 6
The last update that added to the assignment is
the one that was posted Sunday 2/19/23 21:43 EST. In the Monday
2/20 update, I simply removed the "POSSIBLY NOT COMPLETE YET"
line and some interim instructions that were no longer
needed.
Read the handout "One-to-one and onto: What you are really
doing when you solve equations" (a link on the Miscellaneous
Handouts page). The handout addresses
mistaken reasoning of a type I've been seeing in this class when
students solve a system of several simultaneous equations in
several unknowns. Even though the handout's examples all have just
one equation in one unknown, the same principles apply to systems
with more equations and/or unknowns. This type of mistaken
reasoning is common in lower-level courses (even for solving a
single equation in a single unknown), and is rarely corrected
there, but you've reached the stage at which it needs
to be stamped out!
FYI, my filename for this handout is "logic_of_solving_equations".
That's fundamentally what the handout is about. Like most of my
general handouts, this one is written at a (deceptively) simple
level, to make easily readable even for mathematically inexperienced
students. Don't be fooled by the level of the writing. Most
students, even in Real Analysis and Advanced Calculus or beyond,
have never thought about the logic addressed in this
handout. (I suspect that this is one reason that students
often write some portion of a proof as
"Equation. Equation. Equation. ... Equation", with no logical
connectors.)
The "reversibility" illustrated in the
handout's Example 3 (and deciding whether a step is reversible) is
much less clear when we're dealing with systems of more
than one equation in more than one unknown, than when we're
dealing with one equation in one unknown. Later in this
course we will establish the reversibility of certain
steps in solving systems of linear equations. You saw a preview
of this in Section 1.4, but we have NOT proven (or even
stated) the relevant theorems yet.
Before the Mon. 2/20/23 class, finish
reading Section 2.1.
Note that there is actual work for you
to do when reading many of the examples. For those that start with
wording like "Let \({\sf T}: {\rm (given\ vector\ space)}\to {\rm
(given\ vector\ space)}\) be the linear transformation defined by
...", the first thing you should do (before proceeding to the
sentence after the one in which \({\sf T}\) is defined) is to
check that \({\sf T}\) is, in fact, linear. (Example 11
is one of these.) Re-examine the examples
you've already read, and apply the above instructions
to these as well.
Some students will be able to do these
linearity-checks mentally, almost instantaneously or in a
matter of seconds. Others will have to write out the criteria
for linearity and explicitly do the calculations needed to
check it. After doing enough linearity-checks—how many varies
from person to person—students in the latter category
will gradually move into the former category, developing a
sense for what types of formulas lead to linear maps.
In Section 2.1, observe that Example 1 is
the only example in which the authors go through
the details of showing that the function under consideration is
linear. In the remaining examples, the authors assume
that all students can, and therefore will,
check the asserted linearity on their own.
In math textbooks at this level and
above, it's standard to leave instructions of this
sort implicit.
The authors
assume that you're motivated by
a deep desire to understand; that you're
someone who always wants to know
why things are true. Therefore it's assumed
that, absent instructions to the contrary,
you'll never just take the author's word for
something that you have the ability to check;
that your mindset will NOT be (for
example), "I figured that if the book said object
X has property Y at the beginning of an example, we
could just assume object X
has property Y."
2.1/ 1–6, 8, 9, 10–12, 14ac, 15–18,
20. Some comments on these exercises:
-
In 2–6, one thing you're asked to determine is whether
the given linear transformation \( {\sf T:V}\to {\sf W}\) is
onto. In all of these \(\dim(V)\leq \dim(W)\), which makes these
questions easier to answer, for the following reasons:
- If \(\dim({\sf V})<\dim({\sf W})
\), then \({\sf T}\) cannot be onto; see exercise 17a.
- When \(\dim(V)=\dim(W)\), we may be
able to show directly whether \({\sf T}\) is onto, but if not,
we can make use of Theorem 2.5 (when \(\dim(V)=\dim(W)\), the map\({\sf T}\)
is onto iff \({\sf T}\) is one-to-one). We can determine whether
\({\sf T}\) is one-to-one using Theorem 2.4.
Also, regarding the "verify the Dimension Theorem" part of the
instructions: You're not verifying the
truth of the Dimension Theorem; it's a
theorem. What you're being asked to do is to check
that your answers for the nullity and rank satisfy
the equation in Theorem 2.3. In other words, you're doing
a consistency check on those answers.
- In #10: For the "Is \({\sf T}\) one-to-one?" part, you'll
want to use Theorem 2.4, but there's more than one way of
setting up to use it. You should be able to do this
problem in your head (i.e. without need for pencil and
paper) by using Theorem 2.2, then Theorem 2.3, then Theorem
2.4.
-
In 14a, the meaning of "\({\sf T}\)
carries linearly independent subsets of \( {\sf V} \)
onto linearly independent subsets of \( {\sf W} \)" is: if
\(A\subseteq {\sf V}\) is linearly independent, then so is
\({\sf T}(A)\). For the notation "\({\sf T}(A)\)", see the note
about #20 below.
- Regarding the meaning of \({\sf T(V_1)}\) in #20:
Given any
function \(f:X\to Y\) and subset \(A\subseteq X\), the notation
"\(f(A)\)" means the set \( \{f(x): x\in A\} \). (You should already
know this from Assignment 0; it's in the first paragraph of FIS Appendix
B.)
The set \(f(A)\) is
called the image of \(A\) under \(f\).
For a linear transformation \({\sf T}:{\sf
V}\to {\sf W}\), this notation gives us a second notation for the
range: \({\sf R(T)}={\sf T(V)}\).
Read the document "hw_sol'ns_misc_2023-02-19.pdf" that I've posted
in Canvas, under Files.
In the same location, I've also posted a
cover-page from an exam last semester. Familiarize yourself
with the instructions on this page; the cover-page for your
exam will be similar.
In view of the Feb. 22 exam, no
homework will be collected for Assignment 6.
|
| W 2/22/23 |
First midterm exam
In Canvas, under Files, I've posted a
cover-page from an exam last semester. Familiarize yourself
with the instructions on this page; the cover-page for your
exam will be similar.
"Fair game" material for this exam is everything
we've covered (in class, homework, or the relevant pages of the
book) up through Section 2.1
In Chapter 1, we did not cover Section 1.7 or
the Lagrange Interpolation Formula subsection of Section 1.6. You
should regard everything else in Chapter 1 as having been covered
(except that the only field of scalars we've used, and that I'm
holding you responsible for at this time, is \(\bf R\).)
For this exam, and any other, the amount
of material you're responsible for is far more than could be tested
in an hour (or even two hours). Part of my job is to get you to
study all the material, whether or not I think it's going to
end up on an exam, so I generally will not answer questions like
"Might we have to do such-and-such on the exam?" or "Which topics
should I focus on the most when I'm studying?"
If you've been responsibly
doing all the assigned homework, and regularly going
through your notes to fill in any gaps in what you understood
in class, then studying for this exam should be a matter
of reviewing, not crash-learning. (Ideally, this should
be true of any exam you take; it will be true of all of
mine.) Your review should have three components: review your
class notes; review the relevant material in the textbook and
in any handouts (including solutions) I've given; and review
the homework.
When reviewing homework that's been graded
and returned to you, make sure you understand any comments that
Andres or I made on what you handed in, even on problems for
which you received full credit. There are numerous mistakes
that students made for which no points were deducted in homework,
and that Andres and/or I commented on, that could cost you points
on an exam. As the semester moves along, you are expected to
learn from past mistakes, and not continue to make them.
Mistakes that Andres and I have been correcting all semester long are likely
to be
penalized on the exam, some of them heavily (especially if they are
anything addressed in handouts I've assigned you to read).
Note: Failure to pick up your corrected homework,
after being absent when I returned it in class, does not excuse
ignorance of what mistakes of yours have been
commented on or corrected, or ignorance of what penalty-points may have
been assessed if and when you did not follow the
homework submission rules
that have been posted on this page since before the semester began.
|
| T 2/28/23 |
Assignment 7
2.1/ 21, 23, 25, 27, 28,
36. See comments below before starting exercises 23, 25, 28, and
36.
- Regarding #23: The hint for #23 refers you to Exercise 22, an
exercise I would have assigned if not for the fact that we'll be
doing it in class very soon, probably before this assignment is
due. (Actually what we'll be proving in class is the
generalization of #22 to linear maps from \({\bf R}^n\
\mbox{to}\ {\bf R}^m\). You don't need to have done #22 in order
to use the hint, so don't postpone doing #23.
- Regarding #25:
In the definition at the bottom of p. 76, the terminology I
use most often for the function \({\sf T}\) is
the projection [or
projection map] from \({\sf V}\) onto \({\sf
W}_1\). There's nothing wrong with using "on" instead of
"onto", but this map \({\sf T}\) is onto. I'm not in
the habit of including the "along \({\sf W}_2\)" when I
refer to this projection map, but there is actually good
reason to do it: it reminds you that the projection map
depends on both \({\sf V}\) and \({\sf W}\), which is
what exercise 25 is illustrating.
- Regarding #28(b): If you've done the exercises in order, then
you've already seen such an example.
- Regarding #36: Recall that the definition of "\({\sf V}\) is the
(internal) direct sum of two subspaces \({\sf V_1, V_2}\)" had two
conditions that the pair of subspaces had to satisfy Problem 36
says that, when \({\sf V}\) is finite-dimensional and the subspaces are the
range and null space of a linear map, each of these conditions
implies the other. Consequently, for a linear map \({\sf T}: V\to
V\), where \(V\) is a finite-dimensional vector space, you only have to
verify one of these conditions in order to conclude that
\({\sf
V=R(T)\oplus N(T)}\). This is reminiscent of a couple of other
instances of "things with two conditions" for which, under some
hypothesis,
each of the conditions implied the other:
- A set of \(S\) of \(n\) vectors in an \(n\)-dimensional vector
space \({\sf V}\) is linearly independent if and only if \(S\)
spans \(V\). Hence \(S\) is a basis of \({\sf V}\) if either
condition is satisfied.
- Given two vector spaces \({\sf V}, {\sf W}\)
of equal (finite) dimension, a linear map \({\sf T: V\to W}\) is
one-to-one if and only if \({\sf T}\) is onto.
Before Friday's class (2/24), read Section 2.2 up through the
bottom
of p. 80. Before Monday's class (2/27), finish reading Section 2.2.
2.2/ 1–8,12, 16a, (modified as below), 17 (modified as below)
- In #16a: Show also (not instead of) that
an equivalent definition of \({\sf S}^0\) is: \({\sf S^0=
\{ T\in {\mathcal L}(V,W): N(T)\supseteq {\rm span}(S)\}} \).
- In #17: Assume that \({\sf V}\) and
\({\sf W}\) have finite, positive dimension
(see note below).
Also,
extend the second sentence so
that it ends with "... such that \([{\sf T}]_\beta^\gamma\)
is a diagonal matrix, each of whose diagonal entries is either 1 or
0."
(This should actually make the problem easier!)
Additionally, show that if \({\sf T}\) is
one-to-one, then if the bases \(\beta,\gamma\) are chosen as above,
none of the diagonal entries of \([{\sf T}]_\beta^\gamma\)is 0.
(Hence they are all 1, and \([{\sf T}]_\beta^\gamma\) is the \(n\times
n\) identity matrix \(I_n\) defined on p. 82, where
\(n=\dim(V)=\dim(W)\).)
Note: Using a phrase like "for positive [something]"
does not imply that that thing could sometimes be negative! For
example,
as you've already seen in class, "positive
dimension" means "nonzero dimension"; there's no such thing as
"negative dimension". For quantities Q that can be greater than or
equal to zero, when we don't want to talk about the case Q=0 we
frequently say something like "for positive Q", rather than "for
nonzero Q".
On Tuesday 2/28, hand in only the
following problems:
- 2.1/ 25, 27ab, 36
- 2.2/ 2be, 4, 5c, 17 (modified as above)
| T 3/7/23 |
Assignment 8
Exam follow-through:
- Read the exam-solutions
handout posted under Files in Canvas. Make sure you read the
comments in the handout, not just the solutions.
- Go over your exam. Make sure
you understand every comment I made.
- After you've completed the two tasks above, wait a
day or two. Then, without looking at the book or my solutions
handout,
re-do all the exam problems that you didn't get full credit for on
the exam.
This part of the assignment has
several purposes. One, of course, is that you learn the
material you didn't know when you took the exam. Another
is that this task is a self-test of how effective your
mathematical reading is. If, in a no-pressure situation,
you're struggling to do problems that you read solutions
of a day or two ago, that's a sign that your reading did
not accomplish what it should have. Try to figure
out why that happened (which may be different for
different students). It may be that you weren't
sufficiently engaged with the reading.
Reading math isn't like reading anything else.
In order to understand and learn from what you're reading, you often
have to pause to digest the sentence you just read. Always have
pencil and paper (or your preferred substitutes) with you; you may
need to do some "thinking on paper" to convince yourself of something
before returning to reading.
In Section 2.3, read up through Example 2 before the
Wed. 3/1 class. Read up through Theorem 2.16
before the Fri. 3/3 class. (Read the paragraph after
the proof of Theorem 3.16 as well.) Some comments relating
what I did in class to what's in the book (partly in Section
2.2, partly in Section 2.3):
- I didn't get to the notation for composition of linear maps
that's introduced in the first paragraph on Section 2.3 ( \({\sf UT :=
\sf U\circ \sf T}\) ), but learn the notation; I'll use it in the
future.
- I proved Theorem 2.14 in class earlier in the week, or late
last week.
- For general vector spaces \({\sf V,W}\) of finite, positive
dimension $n$ and $m$ respectively, and ordered bases \(\beta,
\gamma\) respectively, on Monday or Wednesday I proved
Proposition M (temporary name, just
for reference below). The map from
\({\mathcal L}({\sf V,W})\) to
\(M_{m\times n}({\bf R})\) given by
\(T\mapsto [T]_\beta^\gamma\) is linear, one-to-one, and
onto.
- Theorem 2.8 is the linearity part of Proposition M. For the special
case \(V={\bf R}^n, W={\bf R}^m\), this linearity is repeated as
Theorem 2.15(c), and the "one-to-one" and "onto" parts of Proposition
M are repeated as Theorem 2.15(d). I derived Theorem 2.15(a) from
Theorem 2.14 (given the definition of \(L_A\),
which I had already stated). In the presence of
part (a), Theorem 2.15(b) is another restatement of the "one-to-one"
part of Proposition M.
- I proved Theorem 2.15(e) on Friday directly. The only part of
Theorem 2.15 that I did not touch on in class is part (f).
- I covered Theorem 2.12 parts (a) and (b) (with a quick verbal
argument), but neglected to do state part (c). Part (c) can be proven
directly, as in the book, or indirectly by using the fact that
\(I_{m\times m}\) and \(I_{n\times n}\) are the matrices of the
identity transformation of \( I_{{\bf R}^n}\) and \(I_{{\bf R}^m}\)
respectively, and the fact that for any sets \(X,Y\) and function
\(f:X\to Y\), the relations \(I_Y\circ f = f=f\circ I_X\) hold.
(Hopefully you covered this in Sets Logic. If not, prove it yourself.)
- After Theorem 2.10, the book mentions that "a more general
result holds for linear transformations that have domains unequal to
their codomains." I sketched a quick verbal proof of that
more-general result, but only for the case in which the vector spaces
involved are \({\bf R}^n, {\bf R}^m,\) and \({\bf R}^p\). (You'll
still need to do Exercise 8 to get the correpond results more
generally.) Also I omitted part (c), but no proof for part (c) is
really needed something that only needs to be stated, because
(again) for any set \(X\) and function
\(f:X \to X\), we have \(f\circ
I_X = f=I_X\circ f\) (a special case of the fact I mentioned at the
end of the previous bullet-point).
- I proved Theorem 2.11 only for the special case
\({\sf V}={\bf R}^n, {\sf W}={\bf R}^m,\) and \( {\sf Z}={\bf
R}^p\).
- Some notation I didn't get to cover is the notation for
"powers"
of a linear transformation \({\sf T}:V]to V\) (defined on p. 98)
or an \(n\times n\)
matrix
\(A\) (see p. 101).
2.3/ 1, 2, 4–6, 8, 11–14, 16a, 17–19.
Some notes on these problems:
- In 1e, it's implicitly assumed that \(W=V\);otherwise the
transformation \({\sf T}^2\) isn't defined. Similarly, in 1f and 1h,
\(A\) is implicitily assumed to be a square matrix; otherwise \(A^2\)
isn't defined. In 1(i), the matrices \(A\) and \(B\) are implicitly
assumed to be of the same size (the same "\(m\times n\)"); otherwise
\(A+B\) isn't defined.
- In 2a, make sure you compute
\( (AB)D\) *AND*
\(A(BD)\) as the parentheses indicate.
DO NOT
USE ASSOCIATIVITY OF MATRIX-MULTIPLICATION IN
THIS EXERCISE. The whole purpose of exercise 2 is for you to
practice doing matrix-multiplication, not to
practice using properties of matrix-multiplication. If your
computations are all correct, you'll wind up with the same answer for
\(A(BD)\) as for \((AB)D\). But, in this exercise, use this
foreknowledge only as a consistency check on your computations,
not as a way to avoid doing computations.
- In #11, \({\sf T}_0\) is the book's notation for the zero
linear transformation (also called "zero map") from any vector
space \(V\) to any any vector space
\(W\). [Conveniently for anyone
who's forgotten where the book introduces this
notation, a reminder appears a few lines earlier in Exercise 9.
You'll also find it on the last page of the book (at least in
the hardcover 5th edition) under "List of Notation, (continued)".
which is the last page of the 5th edition hardcover book. The
book's original definition of the notation seems to be buried in
Section 2.1, Example 8, but you may also remember that you saw
it used on first paragraph of p. 82.]
- In #14, you might wonder, "Why are they defining \(z\) to be
\((a_1, a_2, \dots, a_p)^t\) instead of just writing
\(z=\left( \begin{array}{c}a_1\\ a_2\\ \vdots \\ a_p\end{array}\right)
\) ?" I can't be sure why that was done here,
but historically, this sort of thing was required by
publishers, because it was harder to typeset a column vector than
a row vector, and because the column vector used more vertical space,
hence more paper. It also takes more work in LaTeX to
format a column vector, and it's also a little jarring to see a large column
vector in the middle of line of text.
In the "Convex Sets in Vector Spaces" handout linked to the
Miscellaneous Handout page, read from the beginning up through
Exercise 5 at the top of p. 2, and do Exercises 1–5.
On Tuesday 3/7, hand in only the
following problems:
- 2.3/ 4b, 11, 12c, 13, 16a, 17,
18
- "Convex Sets" handout Exercises 5bc
|
T 3/21/23 |
Assignment 9
Read Section 2.4 before the Friday, Mar. 10 class.
You may skip Example 5, since we skipped the Lagrange Interpolation
Formula in Section 1.6.
The final conclusion
of this example---that \(P_3({\bf R})\cong M_{2\times
2}({\bf R})\)---isn't actually the main point of the example.
We've known since Section 1.6 that \(\dim(P_n({\bf R}))=n+1\) and
that \(\dim(M_{m\times n}({\bf R}))=mn\). Hence, using Theorem
2.19, it follows immediately that \(P_3({\bf R})\cong M_{2\times
2}({\bf R})\) since both of these vector spaces have dimension
4. The book presented Example 5 before Theorem 2.19,
so this easy way of showing \(P_3({\bf R})\cong M_{2\times
2}({\bf R})\) wasn't available yet. However, even without
Theorem 2.19, you (the student) should
easily be able to write down an explicit isomorphism from
\(P_3({\bf R})\) to \(M_{2\times 2}({\bf R})\) (without using
the Lagrange Interpolation Formula), thereby showing
another way that these spaces are isomorphic.
I think the authors' main intent in
Example 5 was to illustrate uses of the tools the book's
argument relies on. If all you want to do is show
that the spaces
\(P_3({\bf R})\) and \(M_{2\times
2}({\bf R})\) are isomorphic, you'd have to be crazy to do it Example
5's way.
BTW: The reason we skipped the Lagrange
Interpolation Formula is time. It's not something that's
fundamental to this course. However, it's actually a very elegant and
beautiful result that addresses a very natural question: given points
\(n+1\) distinct points \(x_1, \dots, x_{n+1}\) on the real line, and
\(n+1\) real numbers \(y_1, \dots, y_{n+1}\), is there a polynomial
\(p\) of degree at most \(n\) such that \(p(x_i)=y_i, \ 1\leq i\leq
n+1\)? There are several indirect ways of showing
that the answer is yes. But the Lagrange Interpolation Formula answers
the question directly, giving an explicit formula for the
unique such polynomial in terms of the data \(
\{(x_i,y_i)\}_{i=1}^{n+1}\). If you're interested, see the "The
Lagrange Interpolation Formula" subsection of Section 1.6 (but be
aware that almost all the notation is different from what I just
used).
2.4/ 1– 9, 13–15, 17,
20, 23. In #2, keep Theorem 2.19 in mind to save yourself a
lot of work. Regarding #8: we did most of this in class, but re-do it
all to cement the ideas in your mind.
Do these non-book problems.
Read Section 2.5.
In the "Convex Sets in Vector Spaces" handout linked to the
Miscellaneous Handout page, read from where you left off up through
the paragraph beginning "Note that in Definition 4, ..."
on p. 4, and do Exercises 6–10.
For students who know some abstract algebra: a
vector space is, among other things, an abelian group (with "+"
being the group operation, and the zero vector being the group
identity element). Subspaces of a vector space are (special)
subgroups. Translates of a subspace \(H\) are what we
call \(H\)-cosets in group theory. (Since the group is
abelian, we need not say "left coset" or "right coset"; they're
the same thing.)
On Tuesday 3/21, hand in only the
following problems:
- 2.4/ 7b (prove your answer), 9b, 14, 15 (just the "only
if" direction). Note: the "only if" direction on #15 is
the direction that says, "If \({\sf T}\) is an isomorphism, then
\({\sf T}(\beta)\) is a basis for \({\sf W}\)."
- non-book problem NB 9.1
- "Convex Sets" handout Exercises 6b, 7, 8b. (Label
these exercises with a "CS" prefix.)
| |
F 3/24/23 |
Special midterm-deal assignment (optional)
This is an opportunity for you to make up some of the points you lost on
Exam 1. Details of the assignment are posted in Canvas, under Files.
|
T 3/28/23 |
Assignment 10
Read the most recent homework-solutions handout
(hw_sol'ns_misc_2023-03-07.pdf) posted in Canvas under Files.
2.5/ 1, 2bd, 4, 5, 6, 8, 11. In 6cd, just find \(Q\), not
\([L_A]_\beta\).
Note that in explicit examples (with
actual numbers), to use the formula
"\([T]_{\beta'}=Q^{-1}[T]_\beta Q\)" to compute
\([T]_{\beta'}\) from \([T]_\beta\) and \(Q\) (assuming the latter
two matrices are known), we need to know how to compute \(Q^{-1}\)
from \(Q\). Efficient methods for computing matrix inverses
aren't discussed until Section 3.2. For this reason, in some of
the Section 2.5 exercises (e.g. 2.5/ 4, 5), the book
simply gives you the relevant matrix inverse. In class, I
showed how to compute inverses of \(2\times 2\) invertible
matrices (modulo my not yet having justified the statement that a
\(2\times 2\) matrix is invertible only if its determinant is
nonzero), so you shouldn't need the book's "gifts" for the
exercises involving \(2\times 2\) matrices. For the exercises
involving \(3\times 3\) matrices, you could figure out \(Q^{-1}\)
by "brute force", computing the \(\beta'\) coordinates of each of
the standard basis vectors \({\bf e_1, e_2, e_3}\) of \({\bf
R}^3\); the \(j^{\rm th}\) column of \(Q^{-1}\) is the coordinate
vector \([{\bf e}_j]_{\beta'}\). However, we will soon have more
systematic, efficient ways of doing this, so I'm sparing you from
doing the extra computation needed to find \(Q^{-1}\) in 6cd.
In the "Convex Sets in Vector Spaces" handout, read from where
you left off up through the end of the examples on p. 5,
and do Exercises 11–15.
In view of the Mar. 29 exam, no
homework will be collected for Assignment 10.
| W 3/29/23 |
Second midterm exam
Location: Little 233
Time: 7:20 p.m.
"Fair game" material for this exam is
everything we've covered (in class, homework, or the relevant pages of
the book and my handouts) up through the Friday Mar. 24 class
and the homework assignment due 3/28/23. The emphasis will be on material
covered since the first midterm.
Re-read the general comments that I posted
on this page in advance of the first midterm (the entry with
the "W 2/22/23" date).
|
T 4/4/23 |
Assignment 11
Do these non-book problems.
3.1/1, 2, 3, 5, 8, 10, 11
(For #5, a proof was sketched in class, but some steps were left to you
as homework.)
On Tuesday 4/4, hand in only the
following problems:
- 3.1/ 2 (just the last part). Display your elementary
row operations using the same notation I used in class (an arrow
pointing from one matrix to the next, accompanied by notation
such as "\(R_1\to R_1+ 3R_4\)").
- Non-book problems NB 11.2abc, 11.3.
|
T 4/11/23 |
Assignment 12
Read the exam-solutions handout for exam 2, posted in Canvas
under Files.
Read Section 3.2, except for the proof of Theorem 3.7.
Students who've been coming to class will realize that my
route to the results in Section 3.2 (and 3.3 and 3.4) is
different from the book's, and that there's terminology I've used
that's not in the book (column space, row space, column rank, and row
rank). This terminology, which I've always found useful
is not my own; it just happens to be absent
from this textbook. Note that once column rank is defined,
my definition of row rank
is equivalent to: \(\mbox{row-rank}(A) = \mbox{column-rank}(A^t)\).
The first definition in Section 3.2 defines
the rank (without the modifier "column" or "row")
of a matrix \(A\in M_{m\times n}({\bf R})\)
to be the rank of the linear map \(L_A: {\bf R}^n\to {\bf
R}^m\).
Using this definition of \({\rm rank}(A)\), and the definitions of
column rank and row rank given in class, below is a summary of the
most important concepts and results in Section 3.2 that may help you
from getting lost in the weeds when reading the book. We'd already
proven all of these by the end of class on Monday 4/3/23).
- Theorem 3.5 can be restated more simply as:
\({\rm rank}(A)=\mbox{column-rank}(A)\).
- Corollary 2c (p. 158)
can be restated more simply as:
\(\mbox{row-rank}(A) = \mbox{column-rank}(A)\).
- Combining the above restatements of Theorem 3.5 and Corollary 2c,
we obtain this restatement of Corollary 2b:
\(\mbox{rank}(A) = \mbox{row-rank}(A)\).
- Corollary 2a, combined with our second definition of row-rank
above ( \(\mbox{row-rank}(A)=\mbox{column-rank}(A^t) \) ), is then
just another way of saying that \(\mbox{row-rank}(A) =
\mbox{column-rank}(A)\).
- Theorem 3.7ab (combined) is what you were asked to prove in
the previously assigned non-book problem NB 9.1 (for which I posted
a solution in Canvas).
The upshot of Theorem 3.5 and Corollary 2 is that
\( {\rm rank}(A)=\mbox{column-rank}(A)=\mbox{row-rank}(A)
\ \ \ \ (*).\)
Since the rank, column rank, and row rank of a matrix are all equal,
it suffices to have just one term for them all, rank. But since
all three notions are conceptually distinct from each other, I
prefer to define all three and then show they're equal; I
think that this makes the content of Theorem 3.5 and Corollary 2
easier to remember and understand. Friedberg, Insel, and
Spence prefer to define only \(\mbox{rank}(A)\), and show it's equal
to \(\mbox{column-rank}(A)\) and \(\mbox{row-rank}(A)\) without
introducing extra terminology that will become redundant once (*) is proved.
Using the previously assigned non-book problem NB 9.1 and the
definition "\(\mbox{rank}(A)=\mbox{rank}(L_A)\)", prove Theorem
3.7cd without looking at the proof in the book.
After doing this exercise,
you may look at the proof of Theorem 3.7 in the book, but first make
sure you've read the posted solution to NB 9.1. As far as I can tell,
the only reason FIS didn't put Theorem 3.7ab into the section where it
naturally belongs (Section 2.3), either giving the proof I gave or
leaving it as an exercise, is that it would have stolen their thunder
for the way they wanted to prove Theorem 3.7. The FIS proof of this
theorem proceeds by first proving part (a) the same way I did, then
proving (c) [using (a)], then (d) [using (c) and "a matrix and its
transpose have the same rank"], then (b) [using (d)]. But Theorem
3.7ab is an important result with instructive, intuitive proofs that
in no way require matrices, or
anything in the book beyond Theorem 2.9. For my money, the
book's proof of Theorem 3.7(b) is absurdly indirect, gives the false
impression that matrix-rank needs to be defined before proving this
result, and obscures the intuitive reason why the result is
true (namely, linear transformations never increase
dimension).
3.2/ 1–5, 6(a)–(e), 11, 14, 15. In #6, one way to do each part is to introduce bases \(\beta,
\gamma\) for the domain and codomain, and compute the matrix
\([T]_\beta^\gamma\). Remember that the linear map \(T\) is invertible
if and only if the matrix \([T]_\beta^\gamma\) is invertible. (This
holds no matter what bases are chosen, but in this problem,
there's no reason to bother with any bases other than the standard
ones for \(P_2({\bf R})\) and \({\bf R}^3\).) One part of #6 can
actually be done another way very quickly, if you happen to notice
a particular feature of this problem-part, but this feature might
not jump out at you until you start to a compute the relevant matrix.
Read Sections 3.3 and 3.4, minus the application on
pp. 175–178.
3.3/ 1–5, 7–10
3.4/ 1, 2, 7, 9, 10–13
On Tuesday 4/11, hand in only the
following problems:
- 3.2/ 3, 6bd, 8
- 3.3/ 2bdf, 4b, 5 (just with \(n=3\))
- 3.4/ 2f, 7, 9. Do #7 by the method recently discussed in
class.
(For those of you who missed class, this method is also
discussed on p. 191, but the discussion there implicitly includes
superfluous steps. [You don't
need to row-reduce all the way to the RREF of \(A\); you can get the answer from
any REF of \(A\).]) Do #9 similarly, after first choosing a basis
\(\beta\) of
\(M_{2\times 2}({\bf R})\) and writing down the coordinate vectors of
the elements of \(S\) with respect to \(\beta\).
|
T 4/18/23 |
Assignment 13
REMINDER: Except when otherwise
specified, no part of the assigned homework is optional. Not a
single exercise, collected or otherwise. Not a single word of
reading—especially (but not limited to) when the reading
is a handout I've written for students' benefit. For example, if
there is any solutions-handout of mine that you still haven't read
(thoroughly), you have not done homework you were assigned to
do.
Do not make the mistake of thinking that homework I'm not
collecting and grading doesn't need to be done. The amount of
work you need to do to truly learn mathematics is vastly
greater than the amount of work that any teacher or TA has time to
grade. If you don't get the non-hand-in homework done by the due
date, do it afterwards (as immediately as possible).
You are expected to do 100% of the assigned homework. For
exercises, "doing 100%" means putting a serious effort
into every assigned exercise. For reading, "doing 100%"
means reading 100%, with your mind focused solely on the
reading (not multi-tasking, for example).
Seriously attempting every assigned exercise, and
doing 100% of the assigned reading, all in a timely fashion, is
the minimum amount of homework you should be doing just to get
a C . "C" means satisfactory. There is
nothing satisfactory about doing less than 100% of the assigned
reading and exercises.
Read Sections 4.1 and 4.2.
4.1/ 1.
Note: The words
"linear" and "multilinear" do not mean the same thing!
4.2/ 1–3, 5, 8, 11, 23–25, 27
. In
general in Chapter 4 (and maybe in other chapters),
some parts of the true/false set of exercises 4.(n+1)/1
duplicate parts of 4.n/ 1. Do as you please with the
duplicates: either skip them, or use them for extra
practice.
Read Section 4.3 up through the last paragraph before
Theorem 4.9 (this material was covered in class, except for
the [important!] Corollary near the bottom of p. 233); skim the remainder of Section 4.3 (unlees
you have the time and interest to read it in depth). I am
not holding you responsible for the formula in Theorem
4.9. (Cramer's Rule is just this formula, not the
whole theorem. You certainly are responsible for
knowing, and being able to show, that if \(A\) is
invertible, then \(A{\bf x}={\bf b}\) has a unique solution,
namely \(A^{-1}{\bf b}.\))
4.3/ 1(a)–(f), 9–12, 15. (For the odd-\(n\)
case of #11, you should
find that 4.2/25 is a big help. In #15, for the definition of
similar matrices, see p. 116.)
Read Section 4.4, as well as my own
summary of some facts about determinants below.
4.4/ 1, 4ag.
If I were asked to do 4g, I would
probably not choose to expand along the second row
or 4th column. Do you see why? If you were asked to compute
\(\left| \begin{array}{cc}
1 & 2 & 3\\ 0& 0 & 4 \\ 5&6&7\end{array}\right|, \)
which method would you use?
Read Section 5.1 before the Mon. Apr. 17 class.
On Tuesday 4/18, hand in only the
following problems:
- 4.2/ 23, 25, 27
- 4.3/ 9, 10, 11, 12, 15
-----------------------------------------------------------------------
Summary of some facts about determinants
In this summary, every matrix \(A, B, \dots,\) is \( n\times
n\), where \(n\geq 1\) is fixed but arbitrary (except when
examples for \(n=1,2\) or 3 are given.)
- The following are equivalent:
- \({\rm rank}(A)=n\)
- The set of columns of \(A\) is linearly independent.
- The set of columns of \(A\) is a basis of \({\bf R}^n\).
- The set of rows of \(A\) is linearly independent.
- The set of rows of \(A\) is a basis of \({\bf R}^n\).
- \(A\) is invertible.
- \(\det(A)\neq 0.\)
(In our coverage of Chapter 2, we showed that the first
six statements on this list are equivalent;
we have simply added a seventh.)
- \( \det(I)=1\) (where \(I\) is the \(n\times n\) identity matrix)
- \(\det(AB)=\det(A)\, \det(B)\)
- If \(A\) is invertible, then \(\det(A^{-1})=1/\det(A). \)
- \(\det(A)=\det(A^t)\)
- If \(A' \) is a matrix obtained by interchanging exactly two columns
of \(A\)
or exactly two rows of \(A\), then \(\det(A')=-\det(A)\).
- If \(A'\) is a matrix obtained from \(A\) by multiplying exactly
one column or row of \(A\) by a nonzero real number \(c\)
(leaving all other columns or rows of \(A\) unchanged), then
\(\det(A')=c\det(A)\).
- For any nonzero \(c\in{\bf R}\), we identify the sign of \(c\)
(positive or negative) with the corresponding real number \(+1\) or \(-1\).
(Of course, "+1" can be written simply as "1".)
This enables us to write equations involving multiplication by
signs, e.g. "\(c={\rm sign}(c)\,|c|\)."
Every ordered basis \(\beta\) of \({\bf
R}^n\) has a well-defined sign associated with it, called
the orientation of \(\beta\), defined as follows:
If \(\beta=\{v_1, v_2, \dots, v_n\}\) of \({\bf R}^n\), where we
view elements of \({\bf R}^n\) as column vectors, let \(A_{(\beta)}
=\left( \begin{array} {c|c|c|c} v_1 & v_2 & \dots & v_n \end{array}
\right) \), the \(n\times n\) matrix whose \(i^{\rm th}\) column is
\(v_i\), \(1\leq i\leq n\).
(The notation \(A_{(\beta)}\) is introduced here just for this
discussion; it is not permanent or standard.)
Then \(A_{(\beta)}\) is
invertible, so \(\det(A_{(\beta)})\) is not zero, hence is either
positive or negative. Wefine the orientation of \(\beta\)
(denoted \({\mathcal O}(\beta)\) in our textbook) to be \({\rm
sign}(\det(A_{(\beta)}))\in \{+1,-1\}.\) Correspondingly, we say that
the basis \(\beta\) is positively or negatively
oriented. For example, the standard basis of \({\bf R}^n\) is
positively oriented (the corresponding matrix \(A_{(\beta)}\)
is the identity matrix).
With \(\beta\) as above, let \(\beta'=\{-v_1, v_2, v_3, \dots, v_n\}\), the ordered set obtained from \(\beta\) by replacing \(v_1\) with \(-v_1\),
leaving the other vectors unchanged. Then \(\beta'\)
is also a basis of \({\bf R}^n\), and clearly \({\mathcal O}(\beta')
=-{\mathcal O}(\beta)\).
Thus there is a one-to-one correspondence (i.e. a bijection)
between the set of
positively oriented bases of \({\bf R}^n\) and the set
of negatively oriented bases of \({\bf R}^n\).
("Change
\(v_1\) to \(-v_1\)" is not the only one-to-one
correspondence between these sets of bases. Think of some more.)
In this sense, "exactly half" the bases of \({\bf R}^n\) are
positively oriented, and "exactly half" are negatively oriented.
(A
term like "in this sense" is needed here since the phrase "exactly
half of an infinite set" has no clear meaning.)
If we treat elements of \({\bf R}^n\) as row vectors,
and define \(A^{(\beta)}\) to be the matrix whose \(i^{\rm th}\)
row is \(v_i\), then \(A^{(\beta)}\) is the transpose of
\(A_{(\beta)}\). Hence, because of the general fact
"\(\det(A^t)=\det(A)\),"
we obtain exactly the same orientation for
every basis as we did by treating elements of \({\bf R}^n\) as column
vectors.
- Determinants and geometry. There is a notion of
\(n\)-dimensional (Euclidean) volume in \({\bf R}^n\)
(let's just call this "\(n\)-volume") with the property that the
\(n\)-volume of a rectangular box is the product of the \(n\)
edge-lengths. The precise definition of \(n\)-volume for more-general
subsets of \({\bf R}^n\) would require
a very long digression, but for \(n=1, 2\) or 3 it coincides,
respectively, with length, area, and what we are accustomed to
calling volume.
In exercise 12 of the "Convex Sets" notes, (closed)
parallelepiped in \({\bf R}^n\) was defined.
For \(n=1\), a
parallelepiped is an interval of the form \([a,b]\) (where \(a\leq b\));
for \(n=2\), a
parallelepiped is a parallelogram (allowed to be "degenerate"
[see the Convex Sets notes or the textbook]);
for \(n=3\), a
parallelepiped is what you were taught it was in Calculus 3
(but allowed to be degenerate).
For an ordered \(n\)-tuple of vectors
\(\alpha=({\bf a}_1, \dots, {\bf a}_n)\)
in \({\bf R}^n\)
let \(A_{(\alpha)}
=\left( \begin{array} {c|c|c|c} {\bf a}_1 & {\bf a}_2 & \dots &
{\bf a}_n \end{array}
\right) \).
(The only difference between this
and our earlier \(A_{(\beta)}\) is that we are not
requiring the vectors \({\bf a}_i\)
to be distinct, or the set
\( \{ {\bf a}_1, \dots, {\bf a}_n\}\) to be linearly
independent.)
For the parallelepiped \(P=P_{(\alpha)}\) in exercise 12 of the
"Convex Sets" notes, with what we may call "edge vectors" \({\bf
a}_1, \dots, {\bf a}_n\), the determinant of \(A_{(\alpha)}\) and
the volume of \(P_{(\alpha)}\) coincide up to sign. More specifically:
- If \(\alpha\) is linearly independent, then
\(\det(A_{(\alpha)})=
{\mathcal O}(\alpha)\times\)
(\(n\)-volume of \(P_{(\alpha)}\)).
- If \(\alpha\) is linearly dependent, then
\(\det(A_{(\alpha)})= 0 =\)
\(n\)-volume of \(P_{(\alpha)}\).
--------------------------------
The following is NOT HOMEWORK. It is enrichment
for students who know some abstract algebra and have
a genuine interest in mathematics.
There is a non-recursive, explicit formula for \(n\times n\)
determinants. To understand the formula, you need to know
(i) what the symmetric group (or permutation group)
\(S_n\) is, and (ii) what the sign of a permutation is.
The formula is this: if \(A\) is an \(n\times n\) matrix, and
\(a_{i,j}\) denotes the entry of \(A\) in the \(i^{\rm th}\) row and
\(j^{\rm th}\) column (the comma in
"\(a_{i,j}\)" is just to make the equation below more
readable). Then
$$
\det(A)=\sum_{\pi\in S_n} {\rm sign}(\pi)\
a_{1, \pi(1)}\, a_{2,\pi(2)}\, \dots\, a_{n, \pi(n)} \ \ \ \ (*)
$$
(a sum with \(n!\) terms, each of which is a product of \(n\) entries
of \(A\) and a sign). (You're forbidden to
use formula (*) on graded work in this class, since
we're not proving it. The fact that it's true is just an "FYI"
for interested students.)
To use formula (*) to prove certain properties of the
determinant, you need to know a little group theory (not
much) and the fact that the map \({\rm sign}: S_n\to \{\pm 1\}\) is
multiplicative (meaning that \({\rm sign}(\sigma\circ \pi)={\rm
sign}(\sigma)\,{\rm sign}(\pi)\ \) ). With that much knowledge, you
can use formula (*) to give proofs of various other facts by
more-direct means than are in our textbook. For example, when proving
that \(\det(A)=\det(A^t)\) or that \(\det(AB)=\det(A)\det(B)\), there's
no need to use one argument for invertible matrices and another
for non-invertible matrices. Of course, formula (*) itself needs proof
first!
There are even better proofs that
\(\det(AB)=\det(A)\det(B)\), but they require far more advanced
tools.
|
T 4/25/23 |
Assignment 14
4.3/ 21. One way to approach this is to use induction
on the number of rows/columns of the square matrix A.
In the inductive step, expand the determinant along the first column.
For students who missed class Friday 4/14 or Monday 4/17: you
are
required to read all of Section 5.1, and Section 5.2 up through
Example 7 (p. 271)
For students who attended both those classes:
your required reading is (i) Example 2 in Section 5.1 (p. 248), (ii)
the generalization of this example that starts three lines from the
bottom of p. 255, and (iii) the portion of Section 5.2 from the
paragraph preceding Theorem 5.7 (p. 264) through Example 7 (p. 271),
and (iv) the subsection entitled "Direct Sums"
(pp. 273–277). In class, as of the end of
Wednesday's lecture (4/19/23), we've covered everything in Section 5.1
(except for some examples and the definitions in the middle of p. 249,
for which exercise 5.1/13, assigned below, is needed) and everything
in Section 5.2 up through the definitions on p. 264 (except for some
examples). However, I still recommend that you read Section 5.1
and the portion of Section 5.2 that we've already covered. In class on
Wednesday, I also stated Theorem 5.7 near the end of class, but
did not have time to prove it.
5.1/ 1, 2, 3abc, 4abd, 5abcdhi, 7–13, 16,
18, 20. I recommend doing 5hi by directly using the definition
of eigenvector and eigenvalue rather than
by computing the matrix of \({\sf T}\) with respect to a basis
of \(M_{2\times 2}({\bf R})\). (I.e., take a general \(2\times 2\) matrix
\(A=\left(\begin{array}{cc} a & b\\ c& d\end{array}\right)
\neq \left(\begin{array}{cc} 0&0\\ 0&0\end{array}\right)\) and
\(\lambda\in{\bf R}\), set \({\sf T}(A)\) equal to \(\lambda A\),
and see where that leads you.)
5.2/ 1, 2abcdef, 3bf, 7, 10.
For 3f, see my recommendation above for 5.1/ 5hi. In #7, the
expression you're supposed to find has an explicit formula for each of
the four entries of \(A^n\).
In Section 6.1, read the first four paragraphs on p. 330, and
read from the middle of p. 331 (the definition
of a norm on a vector space) through Example 8 on p. 333.
Remember that we are restricting attention to real vector spaces
and inner products (as I instructed early in the semester,
wherever a field F appears in the book, replace F
by \({\bf R}\)),
so anywhere you see a bar that represents
complex-conjugation (e.g. in
\(\bar{c}\)
near the top of p. 332), just mentally erase the bar.
6.1/ 8–13, 15, 16b, 20a (this is usually
called polarization—I've never known where the
terminology
comes from—rather than the "polar identity").
Do these non-book problems.
|
No
homework will be collected for Assignment 14.
But do not make the mistake, again, of doing less than 100% of the
homework, or of putting the homework off until you have too little
time to do it and/or to ask questions about it.
Before the final exam |
Assignment 15
I'm listing here some homework related to
the material we're covering in final two lectures. Obviously none of
the exercises will be collected, but you are still responsible for knowing
this material.
If you've respected the
attendance policy announced in the syllabus
(in which case you should have no more than about
2–3 absences, including excused absences), and if you've
kept up with your homework all semester as advised in
the General information section earlier on this
page, as well as in the
syllabus,
as well as in class, and were reminded of again in Assignment 13, you
should have ample time to assimilate the new material, do the new
homework, and do your general review of everything we've covered this
semester, before the final exam.
If you haven't come to nearly every
class, or haven't kept up with homework as advised, well ...
*****************************
6.1/ 3, 17
Read Section 6.2, with the following exceptions and
modifications:
- It's okay if you skip Theorem 6.3; it's less important
than Corollary 1 on p. 340. I proved Corollary 1 in class directly,
without needing to prove Theorem 6.3 first (or at all).
However, the following identity
(which the book uses implicitly to derive Corollary 1 from
Theorem 6.3) is worth recognizing: given any inner-product
space \( (V, \lb \ , \rb\, )\), and any \(w\in V\) and nonzero
\(v\in V\),
$$
\begin{eqnarray*}
\frac{\lb w, v\rb}{\| v\|^2}v
&=&\lb w, \frac{v}{\|v\|}\rb \, \frac{v}{\| v\|}
\ \ \ (*)\\
\\
&=& \lb w, \hat{v} \rb\, \hat{v}
\ \ \ \ \ \ \mbox{if we write $\hat{v}$ for the unit vector
$\ \frac{v}{\| v\|}$ }\ .
\end{eqnarray*}
$$
- To relate Theorem 6.4 (which defines what the book calls the
Gram-Schmidt
process) to what I called the Gram-Schmidt process
in class:
- What I called \(\{v_1, v_2, \dots, v_n\}\) in class
is (effectively) what the book calls \(\{w_1, w_2, \dots, w_n\}\) in
Theorem 6.4.
- What I called \(\tilde{v}_i\) is what the book calls \(v_i\).
- Theorem 6.4 does not include the step in which I normalized
my \(\tilde{v}_i\), i.e. when I defined what I called
\(\hat{v}_i=\frac{\tilde{v}_i}{\|\tilde{v}_i\|} \) in class.
Thus the book's set \(\{w_1, \dots, w_n\}\) (my \(\tilde{v}_1, \dots,
\tilde{v}_n\) ) is only orthogonal, rather
than orthonormal.
Instead, in every application of the book's Theorem 6.4 (Examples 4,
5, etc.), the book normalizes its vectors \(w_1, \dots, w_n\)
(dividing by their norms) only after they've all been constructed,
instead of doing the normalization of each \(w_i\) right after \(w_i\)
(my \(\tilde{v}_i\) has been constructed. This is equivalent
to what I did in class because of the identity (*) above.
To save time and give you a version of the book's equation (1)
that I think is easier to remember, I normalized the new vectors
after each had been constructed, so that in place of the book's
\( \frac{\lb w_k, v_j\rb}{\| v_j\|^2}v_j\) — which in my
notation
would have been \( \frac{\lb v_k, \tilde{v}_j\rb}{\| \tilde{v}_j\|^2}
\tilde{v}_j\) — I could write the simpler \( \lb v_k,
\hat{v}_j \rb \hat{v}_j \).
- The book assumes only that \(\{w_1, w_2, \dots, w_n\}\)
is a linearly independent set, rather than a basis of \(V\).
However, if we replace my \(V\) by \({\rm span}(\{v_1,\dots, v_n\})\),
we get Theorem 6.4.
- I don't care whether you learn the
terminology "Fourier coefficient" defined on p. 345;
the extra terminology is unnecessary in this course. However,
since the terminology is used in some of the exercises I'm assigning:
Wherever you see this terminology in an exercise, "Fourier
coefficients" (for a vector \(x\), with respect to
an orthonormal basis \(\{v_1, \dots, v_n\}\)
mean the inner products \(\lb x, v_i\rb\) appearing in Theorem 6.5.
Thus the Fourier coefficients of \(x\) with respect to a (finite)
orthnormal basis
are simply the coordinates of \(x\) with respect to that basis.
The formula in Theorem 6.5, which I derived in class, shows how easy
it is to find these coordinates when the basis we're using is
orthonormal.
- Skip everything from the paragraph after the definition
on p. 345 through the end of Example 7 (the borrom of p. 346).
- In class, I defined the orthogonal complement of
a subspace \(H\subseteq V\). In the definition on p. 347,
you'll see the same terminology used for an arbitrary nonempty
subset \(S\subseteq V\). The concept is worth having a name
for, but for a general set \(S\) that's not a subspace, I call this
the orthogonal
space of \(S\), not the orthogonal complement. I don't
like seeing the word complement for something that's not a
complement under any conventional use of this word. In the numbered
exercises that refer to "orthogonal complement" of a set that's not a
subspace, replace "complement" by "space". Here is another exercise:
prove what's asserted after "It is easily seen that" in the sentence
that follows the definition on p. 347. (Any time you see
something like this in a math textbook, you
should automatically do it as an exercise.)
And one more exercise concerning this definition:
Show that, for an arbitrary nonempty subset \(S\subseteq V\)
the orthogonal space of \(S\) is the orthogonal
complement of \({\rm span}(S)\).
6.2/ 1abfg, 2abce, 3, 5, 13c, 14, 17, 19
Here are three non-book problems.
|
T 5/2/23 |
Final Exam
Location: Our usual classroom
Starting time: 3:00 p.m.
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