Math 24
Linear Algebra
Last updated March 05, 2023 15:04:16 EST

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Homework Assigments


Week of January 2 to January 6
Assignments Made on:
Monday:
  • Study:
  • Do:
Wednesday:
  • Study: Read section 1.1 and start section 1.2
  • Do:
    1. (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 $xy>0$.
    2. (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.)
    3. (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 Friday's lecture to work this problem.
Friday:
  • 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.


Week of January 9 to Janary 13
Assignments Made on:
Monday:
  • Study: Read Section 1.3
  • Do: In Section 1.3: 3, 19, 23, 28 (Hint: consider #5), and 30.
Tuesday (x-hour):
  • 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).
Wednesday:
  • Study: Read Section 1.5.
  • Do: 8 ($\mathbf F$ having characteristic $2$ just means $1+1=0$), 9, and 19.
Friday:
  • Study: Start Section 1.6
  • Do: In Section 1.6: 3b and 11.


Week of January 17 to 20
Assignments Made on:
Monday:
  • Study: No Class
  • Do:
Tuesday (x-hour):
  • 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.)
Wednesday:
  • 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.)
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.)


Week of January 23 to 27
Assignments Made on:
Monday:
  • Study: No Lecture
  • Do: Study for the exam.
Wednesday:
  • Study: Preliminary Exam
  • Do: Download and complete the take home portion of the exam. You must upload your solutions by noon on Friday.
Friday:
  • Study: Finish Section 2.3 and start Section 2.4.
  • Do:
    • In Section 2.3: 11, 12, and 13.
    • In Section 2.4: 4 and 6.


Week of January 30 to February 3
Assignments Made on:
Monday:
  • 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.
Wednesday:
  • Study: e 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 don't turn it in.
Friday:
  • 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.


Week of February 6 to 10
Assignments Made on:
Monday:
  • 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 don't 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.
Wednesday:
  • 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.
Friday:
  • 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.)


Week of February 13 to 17
Assignments Made on:
Monday:
  • 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.
Wednesday:
  • 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.
Friday:
  • Study: Finish Section 5.1 and start Section 5.2.
  • Do: In Section 5.1: 9ab and 12.


Week February 20 to 24, 2023
Assignments Made on:
Monday:
  • 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.
Wednesday:
  • 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).
Friday:
  • Study: Read Appendix D (on complex numbers) and Section 6.1
  • Do: In Section 6.1: 4b, 6, 9, 10, 11, and 16a.


Week of February 27 to March 3rd
Assignments Made on:
Monday:
  • Study: We'll 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.
Wednesday:
  • Study: No Lecture Today
  • Do:
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.


Week of March 6 and 7
Assignments Made on:
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, and 13.
Tuesday (x-hour)
  • 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.
Friday:
  • Study:
  • Do:


Dana P. Williams
Last updated March 05, 2023 15:04:16 EST