Math 24
Linear Algebra
Last updated January 28, 2023 12:45:52 EST

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Homework Assigments

Week of January 2 to January 6
 Monday: Study: Do: Wednesday: 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 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
 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
 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
 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.