Monday:
- Study: Read section 1.1 and start section 1.2
- Do:
- (EP 1) Show that in an ordered field $\mathbf F$, we always have $1>0$.
(Start by showing that $(-1)^2=1$.) Conclude that if $x<0$ and
$y<0$, then $xy>0$.
- (EP 2) Is it possible to make the complex numbers $\mathbf C$ into an
ordered field? (If $\mathbf C$ is ordered, then $i$ must be
either postive or negative. Now use EP 1.)
- (EP 3) Show that if $V$ is a vector space over $\mathbf F$ and
$ax=\mathbf 0$,
for some $a\in \mathbf F$ and $x\in V$, then either $a=0$ or
$x=\mathbf 0$. (Here $0$ denotes the zero element of the field
$\mathbf F$ and $\mathbf 0$ is the zero
vector in $V$. I suggest using Theorem 1.2 from the text.) You
many want to wait till after Wednesday's lecture to work this problem.
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Wednesday:
- Study: Read sections 1.2 and start section 1.3 of the
text.
Having
a look at Appendices A and B would be helpful. Also Appendic C
on fields might help with the first lecture and the first homework
assignment. But now we are going to settle down and concentrate
on the fields like the rationals, the reals, or later the complex
numbers. In fact, unless explicitly stated otherwise, from this
point on, if you wish you can assume
that our vector spaces are over either the reals or the complex
numbers.
- Do: In Section 1.2: 16, 18, and 21.
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Friday:
- Study: Read Section 1.3
- Do: In Section 1.3: 3, 19, 23, 28 (Hint: consider #5),
and 30.
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Monday:
- Study: Read section 1.4
- Do: In Section 1.4: 6 (assume $\mathbf F$ does not have
characteristic $2$ here), 14, and 15. You should not turn
in problems 7, 8, and 9, but you should be aware of the results
(and that they are straightforward to prove).
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Wednesday:
- Study: NO LECTURE TODAY
- Do:
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Friday:
- Study: Read Section 1.5.
- Do: 8 ($\mathbf F$ having characteristic $2$ just means
$1+1=0$), 9, 19 and 21. (The solution link for 21 is to the wrong
problem, so you'll have to work this one out yourself.)
- Comment: This is a shorter than usual assignment. Sadly,
next week's assignment will be longer than usual.
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Monday:
- Study: Start Section 1.6
- Do: In Section 1.6: 3b and 11.
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Wednesday:
- Study: Finish Section 1.6 and Start Section 2.1. Note
that we are not covering Section 1.7 and you are not responsible for
it. Please pay particular attention to Examples 2, 3, and 4 in
section 1.6 as they will not be covered in lecture.
- Do:
- In Section 1.6: 14, 17, and 23.
- In Section 2.1: 9abc and 11. (Have a look at 1, but do NOT
turn it in.)
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Thursday (x-hour):
- Study: Finish Section 2.1 and start Section 2.2. There
is a lot of material in these sections and we won't be able
to cover it all in lecture. So a careful reading is advised.
- Do:
- In Section 1.6: 33a.
- In Section 2.1: 17 and 27ab. (I also suggest looking at,
but not turning in, 1, 10, and 12.)
- In Section 2.2: 4, 8, and 10. (Also problem 5 would be good
practice. The answer is in the back of the text.)
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Friday:
- Study: Finish Section 2.2 and start Section 2.3.
- Do: In Section 2.2: 14 and 17. (For 17, consider the
proof of the Dimension Theorem.)
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Monday:
- Study: Finish Section 2.3 and start Section 2.4. The
preliminary exam will cover up to and including Section 2.3.
- Do:
- In Section 2.3: 11, 12, and 13.
- In Section 2.4: 4 and 6.
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Wednesday:
- Study: Finish Sections 2.4 and 2.5.
- Do:
- In Section 2.4: 15.
- In Section 2.5: 7a and 10a. You should work, but do not
turn in, 4 and 5 (the
answers are in the back of the text). You might want to think about
10b.
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Thursday (x-hour):
- Preliminary Exam
- Do: The Takehome is due Friday by 10am sharp.
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Friday:
- Study: We are skipping Sections 2.6 and 2.7. Read
Section 3.1 and start Section 3.2.
- Do: No written assignment today. But you are always
well advised to look carefully at quesiton 1 in Section 3.1 (and in
every section we finish). Especailly have a look at problem 3
(the answers are in the back of the text). Try to work problem
12 using induction, but do not turn it in.
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Monday:
- Study: Finish Section 3.2. There is a lot in this
section. Careful reading and review of the lecture would be wise.
There is a lot of computational work in this section as well as
theory. Be sure to work a selction of the parts of 1, 2, 4, 5, and
6 where answers are provided in the back of the text.
- Do: In Section 3.2: 8, 14, 17, and 21. For 21, notice
that if $\operatorname{rank}(A)=m$, then $L_A$ is onto.
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Wednesday:
- Study: Read Section 3.3. We may not finish all of
Section 3.3 today. We are not covering the Leontief model at the
end of the section.
- Do: In Section 3.3: 5 and 8. You do not have to turn
them in, but it would wise to look at question 1,
2, 3, and 4 where the answers are in back of the text.
Friday:
- Study: Read Section 3.4.
- Do: In Section 3.4: 3 and 10. For 10, see Example 4 in
the text. Do not turn in, but you should look at the parts of 1,
2, and 4 that have answers in the back of the text.
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Monday:
- Study: Skim Section 4.1. We are not covering the
material on the area of a paralleogram. Read Section 4.2 through
Theorem 4.4.
- Do:
- In Section 4.1: 5 and 7. You should also work, but not turn
in, 2 and 3a.
- In Section 4.2: 6, 12, 23, and 26. (Problem 23 requires
Theorem 4.4.)
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Wednesday:
- Study: Finish Section 4.2 and read Section 4.3 up to
but NOT including Theorem 4.9 (Cramer's Rule). We are not covering
Cramer's rule in this course. We are not covering Section 4.5 and
will move on to Section 5.1 on Wednesday.
The review in Section 4.4 could be useful.
- Do:
- In Section 4.2: 25. You should be able to check the
answers to problems 13 to 22 easily using the methods in this
section.
- In Section 4.3: 12, 14, and 15.
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Friday:
- Study: Read Section 5.1.
- Do: In Section 5.1, 4bd (for 4b, the characteristic
polynomial is $p(\lambda)=-(\lambda-1)(\lambda-2)(\lambda-3)$),
10.
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Monday:
- Study: Finish Section 5.1 and start Section 5.2.
- Do: n Section 5.1: 9ab and 12.
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Wednesday:
- Study: Finish Section 5.2. We will not cover
Section 5.3. We will also not cover the subsections in Section 5.2
on systems of differential equations or general direct sums.
We may do a little of Section 5.4 and then it will be on to Section 6.1.
- Do: In Section 5.2: 8, 10, 13, and 14. I
suggest looking at problem 12. The
answer is on the web.
- Study: Our Midterm is tomorrow. I covers through
Section 5.1.
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Friday:
- Study: Read Section 5.4.
- Do: In Section 5.4: 11, 13, 17, and 19. Naturally, it
would be good to have a look at 1, 2, and 6 (at least those parts
that have answers in the back of the book).
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Monday:
- Study: Read Appendix D (on complex numbers) and Section
6.1
- Do: In Section 6.1: 4b, 6, 9, 10, 11, and 16a.
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Wednesday:
- Study: We will finish most of Section 6.2 today. Please
note that we may be skipping parts of Section 6.2--6 as we finish
up. You will only be responsible for what we cover in lecture.
- Do: No new written assignment today. However, it would be wise
to work some of the parts of problem 2 (in Section 6.2) that have
answers in the back of the book. You will certainly need to be
able to use the Gram-Schmidt method down the road.
Friday's assignment,
Homework number 8, will be the last assignment to be turned in.
There will be homework assigned, but it will not be graded. That
does not mean that it won't be covered on the final exam.
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Friday:
- Study: Read Section 6.2 and start Section 6.3. We are
not covering least squares approximations and minimal solutions in
Section 6.3.
- Do: In Section 6.2: 7, 8, 11, 17, and 19c.
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Monday:
- Study: Finish Section 6.3 and 6.4.
Recall that we are no longer collecting homework, but this
material could easily find its way onto the final.
- Do: In Section 6.3: 8, 9, 11, 12, 13, and 15. (We will
use the results of problem 15 in lecture.)
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Wednesday:
- Study: Read the revelant parts of Section 6.5.
- Do:
- In Section 6.4: 2a and 9.
- In Section 6.5: 2abd (in part (d) the characteristic
polynomial is $-(\lambda+2)^2(\lambda-4)$), 3, 10, and 11.
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Friday:
- Study: We're following our own path through a small part
of Section 6.7. Note that Monday, May 29th is a
holiday. Our last class will be Wednesday, May 32st.
- Do: I suggest looking at 3abcd. Note that for 3b, it
is wise to consider the transpose as we did in lecture. The
answers for 3ac are in the text.
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