| Date due |
Assignment |
| M 8/28/23 |
Assignment 0 (just reading, but important to
do before the end of Drop/Add)
Note: Since Drop/Add doesn't end
until Tuesday night, Aug. 29, I haven't made Aug. 29 the
due-date for Assignment 1; I'm making that assignment due the
following Tuesday, Sept. 5. But that means that Assignment 1
will include more than a week's worth of exercises and
reading, all of which will be fair game for a Sept. 5 quiz.
You should start working on Assignment 1 as soon as exercises
or reading start appearing for it.
Read the Class home page,
and Syllabus and course information
handouts.
Read all the information above the assignment-chart 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 9/5/23 |
Assignment 1
Read Section 1.1 (which should be review).
1.1/ 1–3, 6, 7. Note: As I mentioned in
class, 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.
In Section 1.2, read up through at least Example 3 before the
Friday Aug. 25 class,
and read the remainder of this section before the Monday Aug. 28 class.
Remember that whenever the book refers to a general field
\(F\), you may assume \(F=\bfr\) (in this class) unless I say
otherwise.
1.2/
1–4, 7, 8, 10, 12–14, 17–20.
Some things to note for these exercises:
- 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).
Even if two or
more of (VS 1)–(VS 8) fail, you need to mention
only one of failed properties. (However if you identify a
property that fails to hold, you still may find it instructive,
and worthwhile practice, to check whether there are other
properties that also fail to hold; that's what the book is doing
in Examples 6 and 7.) For purposes of this
assignment,in any exercise in which you think that (VS n)
does not hold, I want you to write down an example
justifying this conclusion.
To show that something is a vector
space, on the other hand, you have to show that each of (VS
1)–(VS 8) holds.
- In #14,
"C" is the book's notation for the field 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. 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 (rather than writing out a detailed proof in which
you check each of the properties (VS1)-(VS8)).
- The instructions for 1.2/1 should have said that, in this
exercise, the letters \(a\) and \(b\) represent scalars (real
numbers), while the letters \(x\) and \(y\) represent vectors
(elements of a vector space).
Read Section 1.3.
Note: In
class I said, that my definition of "subspace" is equivalent
to, but different from, the book's. This is true, but I
mis-remembered what the book's definition is (the second
paragraph on p. 17). The book's definition is actually fine,
except that I'd rather the book had used the phrase "when
equipped with the operations" instead of just "with the
operations ...".
The two things that
I meant to say were different but equivalent are actually (i)
my definition of "subspace", and (ii) the characterization of
"subspace" given by Theorem 1.3. (Many textbooks take that
characterization as the definition of
"subspace"; that's the approach that I have a
philosophical difference with.)
To jog your
memories, the definition I gave of "subspace of a vector space
\( V \)" is: a nonempty subset \( W\subseteq V \) that is
closed under addition (condition
(b) in Theorem 1.3) and is closed under scalar
multiplication
(condition
(c) in Theorem 1.3). I pointed out in class
that for,
any subset \(W\subseteq V\) (possibly
empty) that is closed under scalar multiplication, "\(W\) is
nonempty" is equivalent to "\(W\) contains \(0_V\)"
(and hence my definition of "subspace" is equivalent to the
characterization given by Theorem 1.3).
Because of this
equivalence, if we want to show that a subset \(W\) of a
vector space \(V\) is a subspace, and we've shown that \(W\)
is closed under \(V\)'s operations of addition and scalar
multiplication, we are not required to show explicitly that
\(0_V\in W\); it suffices to show that \(W\)
is nonempty. Nonetheless, in practice, the easiest
way to show that a hoped-for subspace \(W\) is nonempty is
often to show that \(W\) contains \(0_V\). Thus the "\(W\)
contains \(0_V\)" requirement in Theorem 1.3 is a
nice practical replacement for the more conceptual
"\(W\) is nonempty." This "\(0_V\in W\)" requirement also
leads to a very simple way to rule out various
subsets as subspaces: if \(W\subseteq V\) does not
contain \(0_V\), then \(W\) cannot be a subspace of
\(V\).
1.3/ 1–10, 13, 15, 18, 22.
In #22, assume \({\bf F}_1={\bf F}_2={\bf
R}\), of course.
Note: Any time I say something in
class like "Exercise" or "check this for homework",
do that work before the next class.
I generally don't list such items explicitly on this homework
page.
|
| T 9/12/23 |
Assignment 2
Please do exercise 1.3/18 (see below) before the
Friday 9/8 class.
I will be using the result of this exercise in
class soon, possibly as early as Friday 9/8.
Because of the lecture
lost to the hurricane, I'll need to move some items from my
original "cover this in class" plan to homework exercises. Some of
these may
need to be done before the usual Tuesday due-date. Exercise 1.3/
18 is the first of these.
Some students who responsibly checked
the homework page earlier Wednesday evening, and saw that no
update had been posted yet, might not think to check the page
again on Thursday, and therefore (foregivably) might not see the
"do before Friday" instruction in time. (Nonetheless,
in-class questions that arise because students haven't looked at
this exercise yet, will defeat the purpose of my moving it to
homework.) In the future, if you don't see a homework update
by the end of a day we've had class, you should check again the
next day.
The items I'll be moving to homework will be ones that you
should most easily be able to do on your own. Please make
sure you do all your homework on time, including
these occasional earlier-due-date exercises. Otherwise you may
ask questions that lead me to spend class time on
something easy, which will make me move to homework
something that will be harder for you to do on your
own.
Remember that you should be doing
out-of-class work for this class, and every other math class,
several days a week (at least three). Creating a
work-schedule for yourself that designates one day a week as
your "math day" is NOT smart, and is NOT good
time-management, no matter how well it works for other classes
or activities, or how well it has worked for you in the past.
Students who don't take this advice
dig themselves quickly into holes too deep to dig themselves out
of later. This has not changed in the nearly 40 years I've
been teaching. Thinking that it will be different this year
would be magical thinking.
1.3/ 18, 19. Exercise 18 gives a time-saving way of checking
whether something is a subspace. Essentially, it allows you to
check closure under addition and closure under multiplication at
the same time. (I'll use this "trick" in class only because it saves
time.
You should still conceptualize the two
closure requirements as two distinct requirements.)
Do the following, in the order listed below.
- Section 1.2 (yes, 1.2) exercise #21.
- 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; there is something else
that's also called direct sum. Both types of direct sum are
discussed and compared in the handout referred to below.)
- 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.
Don't worry about what a
field of characteristic two is;
\(\bfr\) is not 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 which you have to come up with
an idea that nobody has 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 \(W_1+W_2=M_{n\times n}(\bfr)\) 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 \(S\). This
suggests that we might be able to find an actual formula
that produces \(A\) and \(S\) from \(B\). Since the definition
of symmetric and anti-symmetric matrices involves
transposes, 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.
Read Section 1.4, minus Example 1.
In the three procedural steps below "The procedure just
illustrated" on p. 28, the 2nd and 3rd steps shoud 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
(beyond Example 1),
but the authors should still have stated the intended meanings
explicitly.
1.4/ 1, 3abc, 4abc, 5cdegh, 10, 13, 14, 16. In
5cd, the vector space in consideration is \(\bfr^3\); in 5e it's
\(P_3(\bfr)\); in 5gh it's \(M_{2\times 2}(\bfr)\).
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.
Side note: The last result we covered in class on Monday
9/11 is almost the same as the unassigned exercise 1.4/12. In
class, what we showed (modulo steps omitted for time near the
end of class, and which you should treat as mini-exercises!)
is that a
nonempty subset \(W\) of a vector space is a subspace iff
\(\span(W)=W\). But this "iff" statement remains true if we delete
the word "nonempty", since
\(\span(\emptyset)=\{0_V\}\neq\emptyset\), and the empty subset of
\(V\) is not a subspace.
| T 9/19/23 |
Assignment 3
1.4/ 2, 6, 8, 9, 11, 15.
Read Section 1.5
1.5/ 1, 2(a)–2(f), 3–7, 10, 12, 13, 15, 16, 17,
20.
Read this
clarification of something in Friday's lecture.
|
| T 9/26/23 |
Assignment 4
In Section 1.6:
- Before the Wednesday 9/20 class, if you see this
assignment in time, read from the beginning of the section
up through at least the statement of Theorem
1.8. (Feel free to continue reading after that!)
- Before the Friday 9/22 class, read from the beginning
of the section up through and including "An Overview of
Dimension and its Consequences" (p. 50)
- Before the Monday 9/25 class, read up through the end of
the "The Dimension of Subspaces" subsection. (You may skip the
only remaining part of Section 1.6: the subsection on the
Lagrange Interpolation Formula.)
Before starting any new exercises, read the blue summary
below of (most of) the definitions and results covered Section
1.6: Although a few of the exercises from Section 1.6 are
doable based only on what we did in class on Wednesday 9/20,
but that would actually defeat their purpose. You're really
intended to do these problems using not-yet-covered
definitions and results.
We'll cover as many of these
definitions and results as possible in Friday's class (9/22).
But I don't want you to wait till after Friday's
class to start practicing with using these definitions and
results. So, with \(V\) denoting a vector space, below is a brief
summary of the content of the
relevant reading
(minus most of the examples). Eventually you'll need to know
not just all the definitions and results in this summary, but
how to prove these results.
Read
these lecture notes for
the lecture that I didn't give on Friday, Sept. 22. At the time
I'm posting this homework-page update, these notes are not yet
complete. But they're going to take me a while to finish, I
didn't want to wait till they were complete, and then drop a whole
lot of pages on you to read all at once. My
notes supplement what's in the book, covering essentially
the same materially a little differently, so it's okay to read my
notes a bit at a time, as more and more of the missed lecture gets
posted. I'm putting a version date/time at the top of the notes
so you'll be able to tell whether the version is new since the
last one you saw. I'll try to remember to update the date/time on
this homework page whenever I post an updated version of the
notes, even if nothing else in this assignment has changed.
Summary of some of Section 1.6's
highlights.
Below, \(V\) is always a vector space.
- \(V\) is called finite-dimensional if \(V\) has a
finite basis, and infinite-dimensional
otherwise. (We'll see a different but
equivalent definition shortly; the definition I'm giving is the
one that's closer to what the terminology sounds like
it ought to mean.)
- If \(V\) is finite-dimensional, then all bases of \(V\) are
finite and have the same cardinality. This cardinality
is called the dimension
of \(V\) and written \({\rm dim}(V).\)
- \( {\rm dim}({\bf R}^n)=n\) (for \(n>0\)).
\( {\rm dim}(\{ {\bf 0}\} ) =0.\)
\({\rm dim}(M_{m\times n}({\bf R}))=mn.\)
\({\rm dim}(P_n({\bf R}) )=n+1.\)
- Suppose \(V\) is finite-dimensional and let \(n={\rm
dim}(V).\) Then:
- No subset of \(V\) with more than \(n\) elements
can be linearly independent.
- No subset of \(V\) with fewer than \(n\) elements
can span \(V.\) (Thus, combining this fact with
the previous one: no linearly independent subset of \(V\) can
have more elements than a spanning set has.)
- If \(S\) is a subset of \(V\) with exactly \(n\) elements,
then \(S\) is linearly independent if and only if \(S\) spans \(V\).
Hence (under the assumption that \(S\) has exactly
\(n\) elements) the following are equivalent:
- \(S\) is linearly independent.
- \(S\) spans \(V.\)
- \(S\) is a basis of \(V.\)
Thus, given a vector
space \(V\) that we already know has dimension n
(e.g. \({\bf R}^n\)), and a specific set
\(S\) of exactly \(n\) vectors in \(V\), if we wish to check
whether \(S\) is a basis of \(V\) it suffices to check either
that \(S\) is linearly independent or that \(S\)
spans \(V\); we do not have to check both of these
properties of a basis.
- If \(V\) is finite-dimensional, then
every linearly independent subset \(S\subseteq V\) can be
extended to a basis (or already is one). (I.e. \(V\) has a basis
that contains the set
\(S\). The sense of "extend[ing]" here means "throwing in
additional elements of \(V.\)")
- Every finite spanning subset \(S\subseteq V\) contains
a basis. (I.e. some subset of
\(S\) is a basis of \(V\).)
- 1.6/ 1–8, 12, 13, 17, 21, 25 (see below), 29, 33, 34.
Among the
exercises for which various facts in the summary below can (and
should) be used to considerably shorten the amount of work
needed are #4 and #12.
Note: For most short-answer homework
exercises (the only exceptions might be some parts of the
"true/false quizzes" like 1.6/ 1), if I were asking you to hand
in the problem, or were putting the problem on an
exam, you'd be expected to show your
reasoning. So, don't consider yourself done if you
merely guess the right answer!
Note: #25 can be reworded as: 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 \({\rm
dim}(W).\)
Both this wording and 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. So, for the second
half of my alternate wording of #25, I really should 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, the latter wording, while
objectively more sensible, has a drawback when teaching: 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 \(V\oplus_e W\).
|
| T 10/3/23 |
Assignment 5
My lecture notes for the
missed 9/22/23 lecture on Section 1.6 are now complete. These
notes actually cover far more material than I could have presented in
a single lecture. Had I not missed the Friday 9/22 class, I would
have presented as much of this material as possible that day, and as
much as possible of the remainder on Monday 9/25; I would have left
the rest for you to read in FIS. (Material that's in my notes that's
not in FIS—the material on maximal linearly independent sets and
minimal spanning sets—would still have been covered in class, as
actually happened to some extent.)
Read any portion of these
notes you haven't already read. My notes now include a presentation
of the "Replacement Theorem" (Theorem 1.10 in FIS Section 1.6) that
I'm reasonably satisfied with. (There's nothing terribly wrong
with the book's proof; I just don't find that it communicates the
proof-strategy in an obvious way, or the intuition behind the
strategy.) I've written out my own proof twice: once with a lot of
comments, and a diagram I think many of you are likely to find
helpful, and a second time with the comments and diagram
removed (so that, when you're ready, you can read the proof
without being interrupted by the comments).
As of 10/3/23, the theorem itself is Theorem 13
on p. 8 of my notes. (Always make sure you're
using the most updated version of any notes or handout I post, since
the latest version often corrects confusing/misleading typos that were
in the previous version.) The annotated version of my proof is
on pp. 8–10, and the un-annotated version is on p. 11. Use
these two versions in whatever way helps you the most: you can read
one and then the other, in either order, or go back and forth between
them.
But once
you think you understand the proof, you should test yourself
honestly on whether you really do understand it: if you can't at
least read the un-commented proof without looking at the
annotated version (or the book's proof), and feel that you could
reproduce the argument, then you don't understand it. And by "feel
you could reproduce it", I don't mean memorize it; I mean that
you ought to be able to come up with essentially the same proof
without looking at notes. (The latter really applies to every result
we've proven all semester, as well as to whatever we prove in the
future.)
1.6/ 14–16, 18 (note: in #18, \({\sf W}\) is not
finite-dimensional!), 22, 23, 30, 31, 32
(Added Saturday 9/30/23.)
Do these non-book problems.
Read Section 2.1 up through Example 13. before the Monday 10/4 class. As in Section 1.6, there
is a lot of content in Section 2.1, more than in any other
section of Chapter 2.
Note that there is actual work
for you to do when reading many of the examples in Section
2.1. In Section 2.1, Example 1 is essentially
the only example in which the authors go through all 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. Examples
2–4 are preceded by a paragraph asserting that the
transformations in these examples are linear, and saying, "We
leave the proofs of linearity to the reader"—meaning you!
In Example 8, the authors neglected to state
explicitly that the two transformations in the example are
linear—but they are linear, and you should show
this. (It's very easy for these two transformations, but it's still a
good drill in what the definition of linearity is, and how to use it.)
When going through examples such as 9–11 in this section (and
possibly others in later sections of the book) that start with wording
like "Let \({\sf T}: {\rm (given\ vector\ space)}\to {\rm (given\
vector\ space)}\) be the linear transformation defined by ... ," or
"Define a linear transformation \({\sf T}: {\rm (given\ vector\
space)}\to {\rm (given\ vector\ space)}\) by ...", the first
thing you should do is to check that \({\sf T}\) is, in fact,
linear. (You should do this before even proceeding to the
sentence after the one in which \({\sf T}\) is defined.)
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 (or at least closer to it), developing a sense for
what types of formulas lead to linear maps.
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 anything
like, "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/ 1a–1f,
2–9, 12, 14–18, 20.
Notes about some of these exercises:
- For 1f, look at #14a first.
- 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, \(\dim(V)\leq \dim(W)\). This 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 \(\dim(V)=\dim(W)\), a linear map \({\sf T}: {\sf V}\to {\sf
W}\) 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.
- Regarding #7: we proved properties 1, 2, and 4 in class, so
most of this exercise will be review.
- 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.
- In #20, regarding the meaning of \({\sf T(V_1)}\): Given any
function \(f:X\to Y\) and subset \(A\subseteq X\), the notation
"\(f(A)\)" means the set \( \{f(x): x\in A\} \). (If you've
done all your homework, you already
saw 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: \({\sf R(T)}={\sf T(V)}\).
|
W 10/4/23 |
First midterm exam
At the exam, you'll be given a booklet with a
cover page that has instructions, and has the exam-problems and
work-space on subsequent pages. In Canvas, under Files >
exam-related files, I've posted a sample cover-page ("exam cover-page
sample.pdf"). Familiarize yourself with the instructions on this
page; your instructions will be similar or identical. In
the same folder, the file "spring 2023 exam1_probs.pdf" has the
list of problems that were on that exam (no workspace, just the
list).
"Fair game" material for this exam is everything
we've covered (in class, homework, or the relevant pages of the
book) up through the Friday Sept. 29 lecture and
the homework due Oct. 3.
The portion of FIS Section 2.1 that's fair game for this exam
is everything up through
Example 13 (i.e. Theorem 2.6 and beyond are excluded).
In FIS Chapter 1, we did not cover Section 1.7 or
the Lagrange Interpolation Formula subsection of Section 1.6.
(However, homework includes any reading I assigned,
which includes all handouts I've assigned. So, fair-game material
includes my posted lecture notes for Sept. 22,
even though the "maximal linearly independent set" material
overlaps with the book's Section 1.7.) 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: reviewing your
class notes; reviewing the relevant material in the textbook and
in any handouts
I've given; and review the
homework (including any reading not mentioned above).
If you're given an old exam to look at, then of course you
should look at that too, but that's the tip of the
iceberg; it does not replace any of the review-components
above (each of which is more important), and cannot tell you
how prepared you are for your own exam. Again, on any
exam, there's never enough time to test you
on everything you're responsible for; you get tested on
a subset of that material, and you should never assume
that your exam's subset will be largely the same as the old
exam's subset.
When reviewing
work that's been graded and returned to you
(e.g. a quiz), make sure you understand any comments
made by the grader, even on problems for which you
received full credit. There are numerous mistakes
for which you may get away with only a
warning earlier in the semester, but that could cost you points if
you're still making them later in the semester. As the semester
moves along, you are expected to
learn from past mistakes, and not continue to make them.
Mistakes that
have
been addressed in handouts you've been assigned to read may be penalized
heavily.
| |
| T 10/10/23 |
Assignment 6
Read the remainder of Section 2.1.
Read Section 2.2.
2.1/ 1gh, 10–11, 20, 21, 22 (just the first part), 23,
25 (see below), 27, 28 (see below), 36.
- 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.
- 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.
| T 10/17/23 |
Assignment 7
2.2/ 1–7, 12, 16a (modified as below), 17
(modified as below).
My apologies! I thought I
had posted the Section 2.2 exercises, exam1_solutions reading, and
Section 2.3 reading Tuesday night, but all I had done was to save the
updated file to
my laptop. I will be adding to this assignment later tonight, but
didn't
want to further delay posting the existing portion of the assignment.
- 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, and 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
\(n=\dim(V)=\dim(W)\).)
Reminder: Using a phrase
like "for positive [something]" does not imply that that thing has the
potential to be negative! For example, as I mentioned 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 \(Q\)".
Read the exam1_solutions handout
and exam1_comments1 handout posted in Canvas under Files.
The comments-handout is "part 1"; I'm planning to add at least
a part 2. I write these handouts to help you, and they're very
time-consuming to write, so please make sure you read
them.
(That's not actually a request ...)
Read Section 2.3.
2.3/ 2, 4, 5 ,
8, 11–14, 16a, 17, 18
Some notes on the Section 2.3 exercises (including some that
haven't been assigned yet):
- 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.
- As you'll see from your reading, and (probably) soon in class,
matrix multiplication is associatve. However, in 2.3/ 2a, make
sure you compute \( (AB)D\)
*AND* \(A(BD)\)
as the parentheses indicate.
DO NOT ASSUME, OR
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 overlooked where the book
quietly introduced this notation, or where it last appeared, there's a
reminder of what the book means by "\({\sf T}_0\)" a few lines
earlier in Exercise 9. You'll also find the notation defined on
the last page of the book (at least in the hardcover 5th
edition) under "List of Notation, (continued)".
I'd have preferred notation such as \({\sf 0_V^W}\), or at least
\(0_{\rm fcn}\)
or \(0_{\rm map}\), for the zero map from \(V\) to \(W\).
- In #14, you might wonder, "Why are they defining \(z\) to be
\((a_1, a_2, \dots, a_p)^t\) instead of just writing \(z=\left(
\begin{array}{c}a_1\\ a_2\\ \vdots \\ a_p\end{array}\right) \)
?" Historically, publishers required authors to write a column
vector as the transpose of a row vector,
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. I can't be whether those were reasons for the book's choice in
this instance, but it's possible. Other possible reasons are (i) it's
a little jarring to see a tall column vector in the middle of line of
text, and (ii) the fact that in LaTeX (the mathematical
word-processing software used to typeset this book) it takes more
effort
to format a column
vector than a row vector.
- In 2.3/ 18:
-
For a linear
map \(\T\) from some vector space to itself,
the book's recursive definition of \(\T^k\) for \(k\geq 2\) is, to me,
not the natural one when \(k\geq 3\) (but is equivalent to what
I consider to be the natural definition). The book's definition is
''\(\T^k=\T^{k-1}\circ \T,\)'' whereas I find
''\(\T^k=\T\circ\T^{k-1}\)'' to be more natural. (For me, the definition
is, ''After you've done \(\T\) \((k-1)\) times, do it one more
time,'' whereas the book's is, ''Do \(\T\), and then do it \((k-1)\)
more times.'')
For any
linear map \(\T:\V\to \V\), the two recursive definitions are
equivalent because function-composition is associative.
Because of the preceding considerations, and the
relationship between linear transformations and matrices, I also think
of the natural recursive definition of \(A^k\) (for a given square
matrix \(A\)) as being ''\(A^k=A\, A^{k-1}\)'', rather than being
''\(A^k=A^{k-1}\, A.\)''
Of course, using the associativity of function composition
and of matrix multiplication, one can easily show that the book's
definitions of \(\T^k\) and \(A^k\) (with \(\T, A\) as above) are equivalent
to the ones I prefer. (Exercise: Do this. Also show that all
non-negative powers of \(\T\) commute with each other, and that all
non-negative powers of \(A\) commute with each other: \(\T^j \T^k =\T^k
\T^j\) and \(A^j A^k = A^k A^j\) for all \(j,k\geq 0.\))
- The book's definitions ''\(\T^0=I_\V\)'' (for any linear map
\(\T:\V\to \V\)) and ''\(A^0=I_{n\times n}\)'' (for any \(n\times
n\) matrix \(A\)) should be regarded as notation conventions for
this book, not as standard definitions like the definition
of vector space. The authors allude to this implicitly in
their definition of \(\T^0\) (''For convenience, we also define
...'') but neglect to do this in their definition of \(A^0\), where
the "For convenience" wording would actually be more important.
|
| T 10/24/23 |
Assignment 8
2.3/ 1, 7, 15, 19
Read Section 2.4 up through the definitions on p. 103 befor
the Wed. 10/19 class. Read the remainder of Section 2.4, except
for Example 5, before the Fri. 10/21 class. After reading Theorem
2.19, go back and replace Example 5 by an exercise that says,
"\(P_3(\bfr)\) is
isomorphic to \(M_{2\times 2}(\bfr)\)." Although that's the
same conclusion that's reached in Example 5, there are much
easier,
more obvious ways to reach the conclusion than the book
illustrates
in thie example.
The point of the book's Example 5 is not actually the
conclusion "\(P_3(\bfr)\) is isomorphic to \(M_{2\times
2}(\bfr)\);" it's to show that one way you could reach this
conclusion is by using the Lagrange Interpolation Formula,
something we skipped in Section 1.6 and won't be covering. Even
without Theorem 2.19, students 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
that these spaces are isomorphic.
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).
Note: The wording of the last part of Theorem 2.18 is not
proper, but could be made proper by either of the following changes:
- After "Furthermore," insert "in the invertible case,"
(including the comma).
- Replace ". Furthermore," (from the period through the
comma)
with "in which case".
Identical comments apply to Corollaries 1 and 2 on
p. 103.
The problem with the book's wording of these results is that
when
the last sentence begins, the objects being inverted are non known to
be invertible. For example, when the last sentence
Theorem 2.18 (as worded in the book) begins, no assumption the
either the linear transformation \(\T\) nor the matrix
\([\T]_\b^\g\) is invertible.
A statement "If P then Q" means
only that IF we assume P is true, THEN we can conclude that Q is true.
It does NOT constitute, or implicitly include, an assumption that P is
true.
Similarly, a statement "P if and only if Q" asserts only that
IF we assume that either of the statements P and Q is true, THEN we
can conclude that the other is true. It does NOT constitute, or
implicitly include, an assumption that P and Q are true.
No assumption that's introduced with an "if" survives beyond
the end of the sentence with that "if" clause.
For the same reason, when a theorem is stated with a hypothesis of
the form "If P [is true]"—for simplicity, let's say that the
theorem is worded in the form "If P, then Q"— then when we
begin the proof, the assumption that P is true is not in effect;
we'll need say something like "Assume (or Suppose) P" to introduce
that assumption.
However, the same theorem could be worded
(completely equivalently,, as far as the content of the
theorem is concerned) as "Assume P [is true]. Then Q [is true]."
With this wording, the assumption that P is true is in
effect already when the proof begins. (Re-assuming P at the start of
the proof would be improper in this case.)
The mistakes above occur frequently in our textbook (and
elsewhere). Every mathematician (including your professor!) occasionally
makes the mistake of stating a theorem as "If P, then Q" and then
starting the proof with the implicit assumption that P is
true.
2.4/ 2–5, 8, 13, 15, 17 (with 17b modified; see below),
23. Some notes on these exercises:
- In #2, keep Theorem
2.19 in mind to save yourself a lot of
work.
- Regarding 17a: in a previous homework exercise (2.1/ 20),
you already showed that the conclusion holds if \(\T\) is
any linear transformation from \(\V\) to \(\W\); we
don't need \(T\) to be an isomorphism.
- Modify 17b by weakening the assumption on \(\T\) to:
"\(\T:\V\to\W\) is a one-to-one linear
transformation." (So, again, we don't need \(\T\) to be an
isomorphism, but we do need more than just "\(\T\) is linear.")
Read the
and exam1_comments2 handout posted in Canvas under Files.
The comments-handout is "part 2" of my earlier handout.
I'm planning to add
a part 3, and/or to add to this part 2.
Some mistakes covered in these comment-handouts were graded fairly
leniently on the first exam. Expect them to be graded less
leniently on the remaining exams.
|
| T 10/31/23 |
Assignment 9
2.4/ 1, 6, 7, 9, 14, 23
Do these non-book problems.
These problems were updated Thursday night.
New problems NB 9.2 – NB 9.6 were inserted, and the problems
formerly numbered NB 9.2 and higher have been renumbered accordingly.
On Friday, typos in NB 9.7(b) were fixed, and
additional info was added to NB 9.1.
Read Section 2.5.
Note: This textbook often states
very useful results very quietly, often as un-numbered corollaries. One
example of this is the corollary on p. 115, whose proof is one of your
assigned exercises.
2.5/ 1, 2, 4, 5, 6, 8, 11, 12.
Note that in explicit examples (with
actual numbers), to use the formula
"\([T]_{\beta'}=Q^{-1}[T]_\beta Q\)" in order to explicitly
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.
But inverses of \(2\times 2\) matrices arise
in examples so often that you should eventually know the following by heart:
the matrix \(A=\abcd\) is invertible if and only
if \(ad-bc\neq 0\), in which case
$$ \abcd^{-1}= \frac{1}{ad-bc}
\left( \begin{array}{rr} d&-b\\ -c&a \end{array}\right).\ \ \ \ (*)
$$
(You should check, now, that if \(ad-bc\neq 0\), and \(B\) is
the right-hand side of (*), then \(AB=I=BA\) [where \(I=I_{2\times
2}\)],
verifying
the "if" half of the "if and only if"
and the formula for \(A^{-1}\). All that remains to show
is the "only if" half of the "if and only if". You should be able to
show this "only if" on your own already, but I'm leaving that for a
later lecture or exercise.)
Warning: Anything that you think is
"mostly correct" (but not completely correct)
version of (*), is useless.
Don't rely on your memory for this formula.
When you write down what you think is the
inverse \(B\) of a given \(2\times 2\) matrix \(A\), always check
(by doing the matrix-multiplication) either that \(AB=I\) or that
\(BA=I\). (Non-book problem NB 9.6 shows why it's sufficient
to do one of these checks.)
This should take you only a few seconds, so
there's never an excuse for writing down the wrong matrix for
the inverse of an invertible \(2\times 2\) matrix.
Read Section 3.1.
3.1/ 1, 3–8, 10, 11. Some notes on these problems:
- 1(c) is almost a "trick question".
If you get it wrong and wonder why, the relevant operation is of type
3. Note that in the definition of a type 3 operation, there was no
requirement that the scalar be nonzero; that requirement was only
for type 2.
- #6 is the exercise I stated in class as the equation
"\(\big({\rm Rop}_x (A)\big)^t ={\rm Cop}_x(A^t)\)."
- In #7 (proving Theorem 3.1), you can save yourself almost
half the work by (i) first proving the assertion just for
elementary row operations, and then (ii) applying #6
and #5 (along with the fact "\((AB)^t=B^tA^t\) " stated and proven
quietly on p. 89).
- In #8, I don't recommend using the book's hint,
especially after you've just done that labor in #7. I would just
use the result of #7 (Theorem 3.1) and Theorem 3.2. (Observe that
if \(B\) is an \(n\times n\) invertible matrix, and \(C,D\) are
\(n\times p\) matrices for
which \(BC=D\), we have \(B^{-1}D= B^{-1}(BC)=(B^{-1}B)C=IC=C,\)
where \(I=I_{n\times n}\).
[Note how similar this is to the argument
that if \(c,x,y\) are real numbers, with \(c\neq 0\), the relation
\(y=cx\) implies \(x=\frac{1}{c}y = c^{-1}y\).
Multiplying a matrix on the left or right by an invertible
matrix (of the appropriate size) is analogous to dividing by
a nonzero real number. But in the matrix case, we
don't call this operation "division".])
Read the handout "Lists and linear independence"
posted on the Miscellaneous Handouts page.
|
|
| T 11/7/23 |
Assignment 10
Read Section 3.2, except (at your option) the
proof of Theorem 3.7.
Also feel free to skip Corollary
1 on p. 157; it's not important enough to be the best use of
your time.
If you've done all your homework
successfully, you already proved a stronger version of
parts (a) and (b) of Theorem 3.7 in homework problem NB 9.1.
Parts (c) and (d) then
follow from parts (a) and (b), just using the definition
of rank of a matrix.
The version
in the homework problem is stronger than Theorem 3.7(b) since
the homework problem did not assume that the vector space
\(Z\) is finite-dimensional. The method of proof I was trying
to lead you to (in the "Something you may find helpful" part
of the problem-statement) is more fundamental and conceptual,
as well as more general, than the proof in the book.
Homework problem NB 9.1 (or, more weakly, 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, further gives the false impression that
Theorem 3.7 needed to be delayed until after Theorem
3.6 and one of its corollaries (Corollary 2(a), p. 156),
and obscures the intuitive reason why the result is
true (namely, linear transformations never increase
dimension).
Students who've been in class this week
(Oct. 30– Nov. 3) will realize that my route to the
results in
Sections 3.2–3.4 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: \(\mbox{row-rank}(A) =
\mbox{column-rank}(A^t)\).
A note about Theorem 3.6:
The result of Theorem 3.6 is pretty, and it's true that
it can be use to derive various other results quickly.
However, the book greatly overstates the importance of
Theorem 3.6; there are other routes to any important
consequence of this theorem. And, as the authors warn in an
understatement, the proof of this theorem is "tedious to
read". There's a related theorem in Section 3.4 (Theorem 3.14)
that's less pretty but
gives us all the important consequences that the book gets
from Theorem 3.6, and whose proof is a little
shorter. Rather than struggling to read the proof of Theorem
3.6, you'll get much more out of doing enough
examples to convince yourself that you understand why the result is
true, and why you could write out a careful proof (if you had
enough time and paper). That's essentially what the book does
for Theorem 3.14; the authors don't actually write out a proof
the way they do for Theorem 3.6. Instead, the authors outline
a method from which you could figure out a (tedious) proof.
This
is done in an example (not labeled as an example!) on pp. 182–184,
though the example is stated in the context of solving systems of
linear equations rather than just for the relevant matrix operations.
3.2/ 1–3, 5 (the "if it
exists" should have been in parentheses; it applies only to
"the inverse", not to "the rank"),
6(a)–(e), 11, 14, 15,
21, 22.
- 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.
Exercises 21 and 22 can be done very
quickly using results from Assignment 9's non-book
problems. (You figure out which of those problems is/are the
one(s) to use!)
In Section 3.3, read up through Example 1.
I've
removed the rest of Section 3.3 from this assignment;
we'll cover that material after the second exam.
Read Section 3.4 up through Example 2, but skipping
the proof of Theorem 3.15.
In a few places, Section 3.4
uses vocabulary introduced on pages of Section 3.3 that I've removed
from this assignment: homogeneous
and nonhomogeneous systems of equations, and homogeneous
system corresponding to the (usually
non-homogeneous) system \(A x= b\). These terms are defined on
p. 170 and in the first paragraph of
p. 172. (The terminology barely enters in
Section 3.4, and it's not something you need to know for the second
midterm. I'm just telling you where to find it just so you can make
sense of the few sentences in Section 3.4 that use it.)
Note that we proved Theorem 3.16(d) in class on
Wednesday 11/1; it's a special case of part (iii) of the proposition
that I spend most of that period on. We essentially proved parts (a),
(b), and (c) of Theorem 3.16 in class on Friday 11/3; I just didn't
state the results in the exact same way as the book.
For now, you are not responsible for the
corollary just below Theorem 3.16. But for future reference, this
corollary and the first sentence of Theorem 3.16 are not worded
well. The corollary should have been worded: Every matrix has a
unique reduced row-echelon form. The term "the reduced
row-echelon form" does not make sense until after uniqueness is
proven.
Other than to understand what some
assigned exercises are asking you to do, I do not care
whether you know what "Gaussian elimination"
means. I never use the term myself. As far as
I'm concerned, "Gaussian elimination" means
"solving a system of linear equations by (anything that
amounts to) systematic row-reduction," even though that's
imprecise. Any intelligent
teenager who likes playing with equations could
discover "Gaussian elimination" on his/her own. Naming such a
procedure after Gauss, one of the greatest mathematicians
of all time, is like naming finger-painting after
Picasso.
3.4/ 1, 2
|
W 11/8/23 |
Second midterm exam
Review the general comments
(those not related to specific content) posted on
this page for the first midterm exam.
Review the instructions on the cover page of your first
exam. The instructions for the second exam will probably be
identical; any changes would be minor.
"Fair game" material for this exam will be
everything we've covered (in class, homework, or the relevant pages of
the book) up through the Friday Nov. 3 class and
the complete Assignment 10.
The fair-game material includes most of Section 3.4 (the portions
indicated in Assignment 10), but does not include anything from Section 3.3
beyond Example 1.
In Canvas, under "exam-related files", I've
posted
the problems from my
Spring 2023 second exam. The fair-game material for your exam will
be more extensive than it was for the spring exam; at the time of
the spring exam, unlike this semester, we had not gotten significantly into
Chapter 3.
The spring's second exam also included a question that would not
have been fair-game material for the spring's first exam, but
was already fair-game material for your first exam.
|
| T 11/14/23 |
Assignment 11
2.5/9. (The definition of "is similar to" is on p. 116.)
In Section 3.4, read Examples 3 and 4.
3.4/ 7, 9–13
Read Section 3.3, minus the application on
pp. 175–178.
3.3/ 2–5, 7–10
|
| T 11/21/23 |
Assignment 12
Read the second-exam solutions posted in Canvas.
In the syllabus, carefully
re-read the
sections "Some advice on how to do
well"
and "Further general advice (for almost
any math class)."
Your CUMULATIVE final exam is less than a
month away. I strongly recommend that you start reviewing
for it NOW (without falling behind on new material and
homework), especially if there is ANY advice in the syllabus
that you did not take. By "NOW" I do not mean "after
Thanksgiving". There's a lot of new material we'll be covering
after Thanksgiving.
Do these non-book problems
that I intended to be part of Assignment 11, but forgot to upload to
the website in time. (Sorry!)
3.3/ 1
Read Sections 4.1 and 4.2.
Note: In the Wed. Nov. 15 class, we proved that a
\(2\times 2\) matrix \(A\) is invertible if and only if \(\det(A)\neq
0\). If you look back at Assignment 9, you'll see that in equation
(*) in that assignment, the denominator \(ad -bc\) on the right-hand
side is exactly \(\det(A)\) (which was assumed nonzero); thus that
equation can be rewritten as
$$ \abcd^{-1}= \frac{1}{\det(A)}
\left( \begin{array}{rr} d&-b\\ -c&a \end{array}\right),\ \ \ \ (**)
$$
where \(A=\abcd\). If you followed the "You should check, now
..." instruction, then, as stated in Assignment 9, you
proved the "if" half of the above "if and only if", by a
method different from the one I used in class on Nov. 15.
Note that, together, the statement, "\(A\) is
invertible if and only if \(\det(A)\neq 0\)" and equation
(**) above, are exactly the book's Theorem 4.2 (p. 201).
- Read Section 4.3 up through the last paragraph before
Theorem 4.9.
- 4.1/ 1–9
- 4.2/ 1–3, 5, 8, 11, 23–25, 27, 29.
As you may notice when doing 4.2/ 1, in Chapter 4
(and occasionally in other chapters), some parts exercise 4.(n+1)/ 1
duplicate parts of exercise 4.n/ 1. Do as you please with the
duplicates: either skip them, or use them for extra practice.
- Do these additional
non-book problems.
(Updated late Thursday night.)
- Read Section 4.3 up through the last paragraph before
Theorem 4.9; 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/ 1a–1f, 9–12, 15, 19. (For the odd-\(n\)
case of #11, you should
find that 4.2/25 is a big help.)
- Read Section 4.4, as well as my own
summary of some facts about determinants below. (The
title of Section 4.4 is somewhat misleading. The book's "summary"
omits many important facts, and intersperses its summarized facts
with uses of these facts (so that the summarized facts
don't appear in a
single list.) The (unlabeled) examples on
pp. 233–235 are useful, instructive, and definitely worth
reading, but hardly belong in a summary of facts about
determinants.)
4.4/ 1, 4ag.
If I were asked to do 4g, I would
probably not choose to expand along the second row
or fourth 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?
-----------------------------------------------------------------------
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.)
(This is a more inclusive list than what I
presented in class on Friday.)
- 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 \(\det(A^{-1})=1/\det(A). \)
- If \(A\) and \(B\) are similar matrices, then
\(\det(A)=\det(B)\)
(assigned exercise 4.3/
15).
- \(\det(A)=\det(A^t)\)
- Determinant is a multilinear function of the columns
(respectively, rows) of an \(n\times n\)
matrix. (See these
non-book problems.)
I.e. if any \(n-1\) of the columns (resp., rows) of the \(n\times n\)
matrix are held fixed, and we allow the other column (rep., row) to
vary over \(\bfr^n\), and take the determinant of the matrix with this
one variable column (resp., row) and \(n-1\) fixed columns (resp.,
rows), the resulting function from \(\bfr^n\to \bfr\) is linear.
In particular,
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
\(\det(A')=c\det(A)\).
- If \(A' \) is a matrix obtained by interchanging exactly two columns
of \(A\),
or exactly two rows of \(A\), then \(\det(A')=-\det(A)\).
In particular, if \(A\) has two identical rows, or two
identical columns, then \(\det(A)=0\).
- \(\det(A)\) can be computed by a cofactor expansion along any
row or column.
- The determinant of a diagonal matrix, or more generally an
upper-triangular or lower-triangular matrix, is the product of the
diagonal entries (cf. assigned
exercises 4.3/ 9, 19)
- Determinants and orientation
For any nonzero \(c\in{\bf R}\), identify the
identify the sign of \(c\)
(positive or negative) with the corresponding real number \(+1\) or
\(-1\),
so that we can 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 \(\beta=\{v_1, v_2, \dots, v_n\}\) is an ordered basis
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. We define 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 volume.
(For this topic, some terminology needs to be
introduced before the relevant fact can be
stated.)
- (Terminology and temporary notation)
For any \(\va_1,
\dots, \va_n\in \bfr^n\) the parallelepiped determined by the
ordered \(n\)-tuple \(\a:=(\va_1,\dots, \va_n)\) is the
following subset of \(\bfr^n\):
$$
\begin{eqnarray*}
P_\a &:=&\left\{t_1\,\va_1+ t_2\va_2+ \dots + t_n\va_n \ : \
0\leq t_i\leq 1,\ \ \
1\leq i\leq n\right\}\\
&& \ \subseteq\ \span(\{\va_1,\dots,
\va_n\})\ \subseteq\ \bfr^n\ .
\end{eqnarray*}
$$
(The notation \(P_\a\) is introduced here just for this
discussion; it is not permanent or
standard.)
Note that if \(n=1\) and \(a=\va_1\), then \(P_\a\) simply the
closed interval in \(\bfr\) with endpoints \(0\) and \(a\).
For the case \(n=2\), convince yourself that if the list
\(\va_1,\va_2\)
is linearly independent, then \(P_\a\) is a parallelogram
(as depicted in Figure 4.3 [p. 203] of FIS), two
of whose adjacent sides are the line segments from the origin to the tips
of \(\va_1\) and \(\va_2\). Convince yourself also that
if the list \(\va_1, \va_2\) is
linearly dependent, then \(P_\a\) is a line segment or a
single point (the latter happening only in the
extreme case \(\va_1=\va_2= {\bf 0}\)). In the latter two cases we
regard \(P_\a\) as a "degenerate" or "collapsed" parallelogram.
Although a parallelepiped is not a subspace of \(\bfr^n\), we
still have a notion of dimension for parallelepipeds.
Specifically, we define the dimension of \(P_\a\), denoted
\(\dim(P_\a)\), to be the dimension of \(\span(\{\va_1,\dots,
\va_n\})\).
If \(\dim(P_\a) < n \)
(equivalently, if the list
\(\va_1, \dots, \va_n\)
is linearly dependent; also equivalently, if \(P_\a\) lies in a
subspace of \(\bfr^n\) of dimension less than \(n\)), we say
that the parallelepiped is degenerate. If
\(\dim(P_\a)=n\) (equivalently, if the list
\(\va_1, \dots, \va_n\) is a linearly independent [hence an ordered
basis of \(\bfr^n\)]), we say that the parallelepiped is
nondegenerate ("solid'').
- (More terminology and temporary notation)
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
\(n~\mbox{-volume}\) 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 the notion of
\(n\)-volume coincides,
respectively, with length, area, and
(3D)
volume.
For an ordered \(n\)-tuple of vectors in \(\bfr^n\), say
\(\alpha=({\bf a}_1, \dots, {\bf a}_n)\), 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 the "\(A_{(\beta)}\)" used
in our discussion of orientation is that we are not requiring the
list \(\va_1, \dots, \va_n\) to be linearly independent.)
- (Fact)
For the (possibly degenerate) parallelepiped
\(P=P_{(\alpha)}\), the determinant of \(A_{(\alpha)}\) and the
\(n\)-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 \(P_{(\alpha)}\).
|
| T 11/28/23 |
Assignment 13
Read Section 5.1.
I suggest
that, immediately after reading the definition of
"diagonalizable matrix" near the top of p. 246, you read (just)
the last sentence of the Corollary on p. 247. That will
give you more concrete idea of what a diagonalizable matrix is: A
matrix \(A\) is diagonalizable iff there exist an invertible
matrix \(Q\) and a diagonal matrix \(D\) such that
\(A=QDQ^{-1}\). After reading the last sentence of the Corollary,
go back to p. 246 and resume reading where you left off.
Read Section 5.2 up through Example 7 (p. 271).
I suggest that before reading
the statement of
Theorem 5.5, you read the following easier-to-understand special case:
Theorem 5.4\(\frac{1}{2}\) (Nickname: "Eigenvectors to
different eigenvalues are linearly independent") Let
\(T\) be a linear operator on a vector space. Suppose
that \(v_1, \dots, v_k\) are eigenvectors
corresponding to distinct eigenvalues \(\l_1, \dots,
\l_k\) of \(T\),
respectively. (Remember that "distinct"
means \(\l_i\neq \l_j\) whenever \(i\neq
j\).) Then the set \(\{v_1, v_2, \dots,
v_k\}\) is linearly independent.
(I've posted a proof of this; see the next assignment.)
5.1/ 1, 2
|
| T 12/5/23 |
Assignment 14
4.3/ 21. (I should have put this in an earlier assignment,
since—as you know if you've been doing your
homework!—one of the proofs in the reading portion of
Assignment 13 made use of the result.)
Read the handout with the file name
"e-vects_to_distinct_e-vals.pdf", posted in Canvas, under
Files/miscellaneous notes. The handout has a proof of "Theorem
5.4\(\frac{1}{2}\)" and some comments.
Although "Theorem 5.4\(\frac{1}{2}\)" is a special case of
FIS's Theorem 5.5, and the proof I've given occupies more space than
the book's proof of the more general theorem, I think you'll find my
proof easier to read, comprehend, and reproduce, partly because the
notation is much less daunting.
In Section 5.2, read the subsection entitled "Direct Sums"
(pp. 273–277).
Read Section 6.1 except for
the following (excluded) material:
- Examples 4 and 9.
- On p. 330: the first paragraph, the second sentence of the second
paragraph, and everything from "A very important ..." to the end of the
page.
Remember that, in this class, we are always
taking the field \({\bf F}\) to be \(\bfr\) unless I say otherwise
(and so far, the only "otherwise" was in our discussion of
polynomials). Thus, every "overbar"
in Chapter 6 can be erased. The same goes for the word "conjugate".
The only reason I didn't exclude the definition on p. 329 from your reading
is so that whenever you see
"\(A^*\,\)" in the book,
where \(A\) is a matrix, you'll know that (for us) the notation \(A^*\) simply
means \(A^t\).
In Section 6.2, read from the beginning up through
Example 3.
5.1/ 3abc, 4abd, 5abcdhi, 7–12 , 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.)
The wording of 18(d) is a example of
less-than-good
writing. The sentence should have begun with "[F]or \(n>2\),"
not ended with it.
5.2/ 1, 2abcdef, 3bf, 7, 10.
For 3f, see my recommendation above for 5.1/
5hi. In #7, you're supposed to find an explicit
formula for each of the four entries of \(A^n\), as was done for
a different \(2\times 2\) matrix \(A\) in an example in Section 5.2.
6.1/ 1, 3, 8–14, 17, 20a.
Some notes on these problems:
- In 1(c), replace "both components" by "each variable,
with the other held fixed". (Equivalently, replace "linear
in both components" with "bilinear".)
- After 1(h), add part 1(g)\('\): If \(y\) and \(z\) are vectors in
an inner product space \( (V, \lb \cdot, \cdot\rb)\) such that,
for all \(x\in V\) we have \(\lb x,y\rb = \lb x, z\rb\), then
\(y=z\).
- What's done in 20a, expressing the inner product
purely in terms of norms, is usually called
polarization (I've never known where the
terminology comes from) rather than the "polar identity".
Do these non-book problems.
|
Before the final exam |
Assignment 15
(There have been no additions since the Dec. 6
update;
the only change since then is that
I've removed the "possibly not complete yet" line that I forgot
to remove sooner!)
Here is some homework related to
the material
we covered in the final two lectures. Its due-date
is the day of the final exam. (But the earlier you do it, the better.)
In Section 6.2, read from where you left off through the end
of the section, with the following exceptions and
modifications:
- Theorem 6.3 is less important than Corollary 1 on p. 340, so
it's okay if to skip Theorem 6.3 as long as you know how to prove
Corollary 1. (I proved Corollary 1 in class directly, without
needing to prove Theorem 6.3.)
However, the following identity
(which which the book uses
implicitly to derive Corollary 1 from Theorem 6.3, and
which I covered in class) is worth
knowing: given any inner-product space \( (V, \lb
\ , \rb\, )\), and any \(w\in V\) and nonzero \(v\in V\),
$$
\begin{eqnarray*}
\frac{\lb w, v\rb}{\| v\|^2}v
&=&\lb w, \frac{v}{\|v\|}\rb \, \frac{v}{\| v\|}
\ \ \ (*)\\
\\
&=& \lb w, \hat{v} \rb\, \hat{v}
\ \ \ \ \ \ \mbox{if we write $\hat{v}$ for the unit vector
$\ \frac{v}{\| v\|}$ }\ .
\end{eqnarray*}
$$
- Skip everything from the definition on p. 345 through the end
of Example 7 (the bottom of p. 346). The definition that I'm having
you skip defines "Fourier coefficient", extra terminology that is
unnecessary in this course. However, since the terminology is used in
some of the exercises I'm assigning: the Fourier coefficients of a
vector \(x\), with respect to a given orthonormal basis \(\b=\{v_1,
\dots, v_n\}\), are simply the coordinates of \(x\) with respect to
\(\b\). As shown in class on Monday 12/4, these coefficients reduce
simply to the inner products \(\lb x, v_i\rb\) appearing in Theorem
6.5, making them very easy to compute (without solving any
simultaneous equations, row-reducing any matrices, etc.).
- In class, we defined, and proved things about,
the orthogonal complement of a
subspace
of \( (V, \lb \cdot, \cdot\rb) \).
On p. 347, the terminology "orthogonal complement of \(S\)" is
defined (unfortunately) for an arbitrary nonempty
subset \(S\subseteq V\). Although the concept for such general
\(S\) is worth naming,
the term "orthogonal space of \(S\)" is a better name than
"orthogonal complement of \(S\)" when \(S\) is not a subspace.
In the numbered
exercises that refer to "orthogonal complement" of a set that's not a
subspace, replace "complement" by "space"; it's really not good to use
the word complement used for something that doesn't resemble a
complement under any conventional meaning of the word.
A couple of exercises related to this definition:
(1) 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.)
(2) Show that, for an arbitrary nonempty subset \(S\subseteq V\)
the orthogonal space of \(S\) is the orthogonal
complement of \({\rm span}(S)\).
Note: Comparing my in-class presentation of the
Gram-Schmidt process to the one in the book (Theorem 6.4), my \(\{v_1,
v_2, \dots, v_n\}\) is (effectively) the book's \(\{w_1, w_2, \dots,
w_n\}\) and vice-versa. Sorry for the notation-reversal!
The book assumes only that its \(\{w_1, w_2,
\dots, w_n\}\) is a linearly independent set, rather than a
basis of \(V\). However, if we replace my
\(V\) by \({\rm span(my\ }\{v_1,\dots, v_n\})\), and swap \(v\)'s and
\(w\)'s, we get Theorem 6.4.
The book's statement of Theorem 6.4 has a bit
of a problem: equation (1) cannot be written before knowing
that each of the book's vectors \(v_j\) (my \(w_j\)), \(1\leq
j\leq k,\) is nonzero; the sentence after this equation, as written,
comes too late. Fixing this problem requires a lengthier statement of
Theorem 6.4 than the one in the book. That's why I didn't state the
theorem the book's way in class.
6.2/ 1abfg, 2abce, 3, 5, 13c, 14, 17 (remember
that the book's \(\T_0\) is the zero operator), 19
Do these
non-book problems
(updated 12/6/23 5:12 p.m.).
|
Thurs 12/14/23 |
Final Exam
Location: Our usual classroom
Starting time: 10:00 a.m.
|