Discussion thread 🔥

[bestsauce]

Note: this is a discussion thread only about the way we set things up on Github. Discussions on the book are for their relevant threads, as explained in README.

• Common milestones?

• All exercises or divide them into necessary+optional ones?

• Dividing exercises issues into each section instead of each chapter?

• Working all on a single document for notes?

Testing $\LaTeX$: $\displaystyle\int_0^1 \dfrac{x^2}{x+1}\ \mathrm{d}x$.

$$\boxed{\ \oint{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = - \overset{\color{white}{.}}{\frac{\mathrm{d}}{\mathrm{d}t}} \underset{\color{white}{.}}{\iint\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S}\ }$$

[bestsauce]

You're welcome @cpassos ! :) I sent you an invitation, visit https://github.com/worstsauce to accept the invitation. Once done, do you want to start working on chapter 3 (which we work on right now)?

I'm interesting in participating. I have been using > this site> to check my answers, but some are either missing or proven differently. I look forward to post some of my solutions here and see if you guys can point out a mistake I could've possibly made.

— You are receiving this because you were mentioned. Reply to this email directly, > view it on GitHub> , or > mute the thread> .

[celiopassos]

@bestsauce Thanks. Chapter 3 is just the one I'm on right now. How does this work? Everyone has a separate issue with exercises?

[bestsauce]

@cpassos

There are two Chapter x issues, one with the Notes label and the other with Exercises label. As you study along you may add the additional help wanted label to get some help (e.g. hint), or question if it's just a question (e.g. can anyone check if my solution is correct? I don't understand this part of the proof, can anyone clarify it?), and remove them once you're done.

Here are you two issues for chap3: https://github.com/worstsauce/study/issues/20 and https://github.com/worstsauce/study/issues/19

In addition, we still have a question that needs to be answered for chap1 :) https://github.com/worstsauce/study/issues/8

Also here's a short tutorial on how to get MathJax working with Github: https://github.com/worstsauce/study#how-to-mathjax--github

[MaxisJaisi]

@cpassos If you're into writing notes for yourself, you could perhaps share them here. I did it for Chap 1 & Chap 2, but they are incomplete, and devoid of good examples. As you can see from our discussion I'm planning to send my TeXed notes to @bestsauce; he'll work his magic on it, and we'll see what happens.

What's your (academic) background, if you don't mind? Why linear algebra?

Also, I'm guessing, from your name, that you're Italian... ?

[celiopassos]

@MaxisJaisi I'm from Brazil. I just started my major in math. I took interest in Axler's book due to the way it is laid out (everything is detailed and no determinants until the end) and so I could develop my proof writing skills.

[bestsauce]

Well looks like we're stalling again...

[celiopassos]

@bestsauce You guys are, I'm not, I will finish Chapter 3 in 2 weeks. I'm doing section F now, after that I'll redo section E, because I didn't actually go through all of it, and after that I'll just review C and D.

[MaxisJaisi]

@bestsauce

It's interesting to think about why exactly we haven't been keeping up with the schedule. I've seen many online study groups that have failed, and always wondered why. Now that I've been in one, I can only speculate, and these reasons will be personal. I think it may help to reflect on why there was lack of progress:

1. Number of people engaged in the program. There were only two of us, and our interaction had been quite impersonal, so it was difficult to maintain the momentum. There are not enough people to "push" the program forwards, so to speak: it's all too easy to stall when there are only two people, because we could always push the deadline forwards. During the early stages of our group study, I would procrastinate, and then towards the end of the week I'd see @bestsauce doing more problems and asking more questions, and I'd naturally attempt the problems on my own and try to catch up. Without this impetus I found myself working on other things.

2. Being engaged in another demanding activity. I am working through a book on real analysis, and when I get to a difficult portion of the text, I find myself not being able to disengage from thinking about it, and this leads to my neglecting linear algebra. I guess this is personal, because some are better at multitasking than others, but I stubbornly refuse to budge when there's an unresolved problem.

3. It's difficult to start after a period of inactivity. This goes without saying, but I'm finding it very difficult to set my mind straight to think about linear maps between vector spaces after several weeks of abandoning the subject. This is in direct contrast with the momentum I had when I focused all my time on chapter 1 and chapter 2. I think extending deadlines isn't a healthy thing to do... We should at least move forwards, even though we haven't fully mastered previous concepts. We can always go back and fill in the details in the future.

4. "Perfecting notes". I spent way too much time thinking about how to rewrite linear algebra concepts in my own terms, to the extent that I neglected the problem sets. I think I should focus more on the problems from now on, and write the notes only after finishing a chapter. Also, I spent an inordinate amount of time worrying about "getting the perfect picture of the subject" by consulting way too many sources and digging too deeply, which isn't exactly the wisest thing to do, considering the fact that this is my first exposure to the subject.

5. Lack of previous exposure to the subject. This bothered me the most. Although Axler's approach is very aesthetic and clean, I found myself limited by my lack of exposure to elementary computational linear algebraic concepts, like manipulating matrices, solving linear equations, row operations, and the connection between properties of matrices and the maps they represent. I then spent a lot of time practising these things using other sources, to the detriment of my progress in this program.

Anyway, I hope @cpassos will stay, because three is better than two. I'll try my best to catch up in these 2 weeks...

[MaxisJaisi]

@cpassos Again, out of curiosity, are you taking a linear algebra course at uni currently?

[celiopassos]

MaxisJaisi I will definitely stay if you do as well.

In my school, we have just finished a class they call "Analytic Geometry and Linear Algebra". I think it is supposed to just give a geometric intuition about linear algebra concepts, since there was no rigour (we didn't see a single proof in the entire program). Unfortunately, our professor was pretty bad, so we didn't develop intuition either, just made some mindless computations.

[MaxisJaisi]

@cpassos I will. Lack of a proper background in LA cripples me severely when it comes to learning interest and important mathematics (chiefly analysis of several real variables, calculus on manifolds...), so I definitely must finish Axler this year. I just lost momentum, that's all. Will get back to Chap 3 once I tidy up a few things I'm doing in analysis today.

[bestsauce]

@MaxisJaisi I think you can add that we set all exercises to make, which is too demanding (with respect to our schedule). I think we should be more selective.

Anyway, glad we're still alive, going to reset the milestone.

[MaxisJaisi]

@bestsauce Do you agree that we should open an issue for 3.C & 3.D? I'm about halfway through the exercises in 3.B, but also completed sections 3.C & 3.D (though I haven't touched the exercises). Still trying to internalise quotient and dual spaces, but I think I can multitask. Don't close the 3.B issue first.

[celiopassos]

Guys, I'll be posting my solutions only on this repo and not on the issues, because I like to keep each statement on its own line but GitHub renders newlines differently in issues (which make my solutions look weird) and I don't I want to have to keep two versions up to date.

Nevertheless, I will still follow what you guys post here and create issues here to post questions, just not the solutions.

[celiopassos]

Looks like @bestsauce has deserted us.

[MaxisJaisi]

@cpassos Sounds good. Actually the solutions posted on issues are temporary. My solutions were transferred to separate repositories after completing the previous 2 chapters. I have thoughts about going back and cleaning up my solutions after I'm done with the whole book, since in hindsight, I realise that some of my previous solutions were quite drawn out and confusing.

By the way, nearly done with the exercises in 3B, and will get to the problem you asked soon. After finishing 3B I'll jump ahead to 3E & 3F, because I'm least comfortable with these chapters ( still trying to picture quotient spaces... ). And then after I'm done with chapter 3, I'll skip the polynomials chapter and head straight to eigenvectors & eigenvalues. What do you think? If you feel you want to start with chapter 4 and chapter 5, you could do so, I'll trail behind, but it's still possible to cross check our solutions.

Perhaps @bestsauce is busy?

[celiopassos]

@MaxisJaisi I'm also finding quotients spaces rather difficult, I still couldn't do the exercises. Axler says in the preface that this chapter, and the duality one, is a bit abstract and can be skipped without running into problems later in the book. Perhaps we should put them aside for now and come back at them later once we develop more intuition. This strategy works well for me when I find something too difficult. I always discover new insight when I look at something a second time.

About chapter 4, I think going through it is worth it, since it is so short. Also, some of the results there are proved using linear algebra, which is nice.

[MaxisJaisi]

@cpassos Okay, I'll do that --- skip quotient spaces and duality. Will finish up the remaining exercises in the chapter, and then go to polynomials.

[MaxisJaisi]

@cpassos I'll be away for a week (preparing for an entrance exam). Will be back when it ends!

[celiopassos]

@MaxisJaisi Ok.

[MaxisJaisi]

Done with exams.

@cpassos, I just read your answer to Exercise 24, 3.B. It seems that it can be simplified considerably. First, do the same thing as you did (take a preimage set) for $T{1}$, construct the map in the same way, call it $S$. Then the linear map we want is $T{2} \circ S$. We verify that $T{2} \circ S \circ T{1} = T{2}$ by noting that $S(T{1} (v)) - v \in \operatorname{null} (T_{1})$, for all $v \in V$.

Also, you assumed that $\operatorname{null} T{1} = \operatorname{null} T{2}$, but the question only says that one is a subset of the other, not that they are equal.

[celiopassos]

@MaxisJaisi

Then the linear map we want is $T_2 \circ S$.

We want just $T_2$. $T_2 S$ is in exercise 25.

We verify that $T_2 \circ S \circ T_1 = T_2$ by noting that $S(T_1(v)) − v \in \operatorname{null}(T_1)$, for all $v \in V$.

I guess $T_2 \circ S \circ T_1$ doesn't make sense because the domain of $T_2$ is $V$, but the codomain of $S$ is $W$. Perhaps you mean to construct $S: \operatorname{range} T_1 \to V$ such that $Sw_j = v_j$? Could you elaborate this a bit more? I can't see how to prove that part you note without first showing that $V = \operatorname{null} T_1 + \operatorname{span}(v_1, \dots, v_n)$.

Also, you assumed that $\operatorname{null} T_1 = \operatorname{null} T_2$, but the question only says that one is a subset of the other, not that they are equal.

Thanks, I fixed it. This mistake did not influence the rest of the proof though, because no part in it required that $\operatorname{null} T_2 \subset \operatorname{null} T_1$.

[MaxisJaisi]

@cpassos I apologise for the lousy write-up, I typed things up on the fly, and was rather jaded after the exams. Anyway, let me explain it again, hopefully more carefully this time:

I'll paraphrase Axler's question again.

Let $T{1}, T{2}$ be linear maps from $V$ to $W$, where $W$ is finite-dimensional. Then $\operatorname{null} T{1} \subset \operatorname{null} T{2}$ only if there exists a linear operator $S$ on $W$ such that $ST{1} = T{2}$.

The main intuition (I think) is that if $T{1}$ is invertible, then it's easy to construct $S$: We first send everything back, from $W$ to $V$, by applying the inverse of $T{1}$, let's call it $F$, to $T{1}$. We get $F \circ T{1} = e{V}$. Then we easily have, by applying $T{2}$ on the left, $T{2} \circ F \circ T{1} = T_{2}$.

But $T{1}$ isn't invertible, so we have a problem. We get around this problem by constructing a "partial inverse" of $T{1}$: we first specify a basis $w{1}, w{2}, \dots, w{n}$ for $\operatorname{range}(T{1})$. Then there must be a $v{i} \in V$ such that $T{1} (v{i}) = w{i}$, for $1 \leq i \leq n$ (note that this doesn't involve the axiom of choice. in general, it does, if we don't restrict $W$ to be finite-dimensional, but Axler wants to avoid choice). Note also that we're fixing $v{i}$, by specifying a single preimage from the set of preimages for each basis $w{i}$. We build a "left partial inverse" $L$, acting on $W$, as follows: $L(w{i}) = v{i}$ for each $1 \leq i \leq n$, and extend linearly to the entire basis of $W$.

Then I claim that $T{2} \circ L$ will be the $S$ that we want. This is where the hypothesis $\operatorname{null}(T{1}) \subset \operatorname{null}(T{2})$ comes in. To see that $T{2} \circ L \circ T{1} = T{2}$, consider $v \in V$, and we compute:

$S \circ T{1} (v) = T{2} \circ L \circ T{1} (v) = T{2} (L(T{1} (v))) = T{2} (L(a{1} w{1} + a{2} w{2} + \dots + a{n} w{n})) = T{2} (a{1} v{1} + a{2} v{2} + \dots + a{n} v_{n})$.

Now if we can show that $T{2} (a{1} v{1} + a{2} v{2} + \dots + a{n} v{n}) = T{2} (v)$, we are done. The expression is equivalent to $T{2} (a{1} v{1} + a{2} v{2} + \dots + a{n} v{n}) - T{2} (v) = \mathbf{0}$, which is equivalent to $ T{2} (v) - T{2} (a{1} v{1} + a{2} v{2} + \dots + a{n} v{n}) = \mathbf{0}$, which is equivalent to (since $T{2}$ is linear), $ T{2} (v - (a{1} v{1} + a{2} v{2} + \dots + a{n} v{n})) = \mathbf{0}$, which is equivalent to $v - (a{1} v{1} + a{2} v{2} + \dots + a{n} v{n}) \in \operatorname{null} (T{2})$, which is true, because $\operatorname{null}(T{1}) \subset \operatorname{null}(T_{2})$! (I'll leave it to you to verify this last step)

Well, the writeup was rather painful, so let's rewrite it in a more streamlined fashion, which might confuse someone encountering it for the first time:

Proof: We first build a partial inverse, $L$ of $T{1}$ as follows: let $w{1}, \dots, w{n}$ be a basis for $\operatorname{range}(T{1})$ (this enumeration is possible, since $W$ is finite-dimensional.) Given each $w{i}$ ($1 \leq i \leq n$), we pick a corresponding $v{i} \in V$ such that $T{1} (v{i}) = w{i}$. Then define $L$ on $W$ by $L(w{i}) = v_{i}$ ($1 \leq i \leq n$), and extend linearly to a full basis of $W$.

I claim that $S = T{2} \circ L$ works, that is, $T{2} \circ L \circ T{1} = T{2}$. To see this, we first note that for any $v \in V$, we have $L(T{1} (v)) - v \in \operatorname{null} (T{1}) \subset \operatorname{null} (T{2})$, so $T{2} ((L(T{1} (v)) - v) = \mathbf{0}$, which is equivalent to $T{2} \circ L \circ T{1} (v) = T{2} (v)$.

[MaxisJaisi]

@cpassos Strange, the paragraph after the quoted question doesn't render correctly, and I can't seem to work out what went wrong. Anyway, you can ignore it for now, it doesn't affect the rest of the argument, and I'll try fixing it afterwards.

Edit: Fixed

[celiopassos]

Thanks! That is really nice, much shorter than mine and avoids altogether the $V = \operatorname{null} T_1 + \operatorname{span}(v_1, \dots, v_n)$ thing.