My experience with eigenstuff has been kind of a slow burn. At first it feels like “that’s it?”, then you do a bunch of tedious calculations that just kind of suck to do… But as you keep going they keep popping up in ways that lead to some really nice results in my opinion.
Eigenvectors, values, spaces etc are all pretty simple as basic definitions. They just turn out to be essential for the proofs of a lot of nice results in my opinion. Stuff like matrix diagonalization, gram schmidt orthogonalization, polar decomposition, singular value decomposition, pseudoinverses, the spectral theorem, jordan canonical form, rational canonical form, sylvesters law of inertia, a bunch of nice facts about orthogonal and normal operators, some nifty eigenvalue based formulas for the determinant and trace etc.
I’ve been knocking out the trig problems in this section with minimal difficulty so far, but I’ve run straight into a brick wall on this “Algebraic” part. I’m asked to find sin(x)=0 between [0,2π). If I graphed the unit circle this would be a trivial exercise to show sin(θ)=0 when θ=0 or π....
There’s nothing here that tells you that you need to use some technique from algebra explicitly as far as I can see?
Edit: I misread. See my follow up comment too.
The usual trig functions like sine and cosine are famous examples of transcendental functions so I very seriously doubt there is some clever algebra trick you’re missing that the author intends you to do.
I’d assume your instructor (or the author) is expecting you to use your geometric knowledge about the unit circle on these if you haven’t covered inverse trig functions yet. But I also can’t really read their mind, so your best bet might be to just directly ask them?
Edit: This could also depend on what trigonometric identities you know so far too.
The point they made was correct. Arcsine by itself only gives one of the two solutions to sinx=0. It seems like they already realize that they need to use arcsine carefully if they use it at all.
Okay, I see. I’m fucking blind and did not see the words “algebraic” literally at the top of the screenshot.
For what it is worth, they could just be referring to how they are representing the problems they are asking rather than the form of the intended solutions with that.
I realized in retrospect I misread the header so I apologize for that.
I’m still betting they aren’t expecting a true algebraic or analytic solution here. Things like finding max/min points, finding arbitrary particular values of trig functions, solving trigonometric equations and so on can be notoriously hard in the absence of geometric reasoning/intuitions.
Later on if you decide to study calculus you might eventually see the sine and cosine functions defined rigorously via infinite series. That may sound convoluted, but part of the purpose of doing that is because of the difficulties of the issues mentioned above. Basic sounding facts like: What is sin(0.1234)? are not so easy to answer where you are at but can be dealt with more conveniently using these kinds of tools from calculus.
The questions being asked here are also just kind of typical knee jerk facts that most people want students coming out of a trig class to just know.
I think your reasoning geometrically seems very on the right track. Appealing to the unit circle or the graph of y=sinx for these feels correct in the sense of what a trig student would be expected to know coming out of or during a trig course.
Rewriting the problem as solving sin(A)=0 and then claiming outright that A must be an integer multiple of pi doesn’t really help as far as I can tell, since that is just the original problem with x exchanged for A?
My issues with this are: Your solution did not originally claim this, it is not stated anywhere in the problem and it leads to exactly the same kind of foundational issues in the context of showing “algebraically” why cos(x)=1 at integer multiples of 2pi now.
The question as given is illposed. You have to know something. If not, why not ask a philispohical question like what is trigonometry even?
Agreed. It’s at least vague/misleading. This is apparently for a precalc clep exam, so the only real sane definition a student would know to fall back on here would be geometric definitions for sine and cosine. What I think the intent of the problem is, is to build intuition on knee jerk facts about sine/cosine rather than something particularly formal?
“The cardinality of the integers is equal to the cardinality of the real numbers, which is called the continuum hypothesis.”
The cardinality of the integers is not equal to the cardinality of the reals. The integers are countable (have the same cardinality as the natural numbers). A very famous proof in set theory called Cantor’s diagonal argument shows the reals are uncountable (i.e. not countable).
The continuum hypothesis is also not about comparing the cardinality of the reals and the integers or naturals (since we already know the above). The continuum hypothesis is about comparing the cardinality of the reals with aleph_1.
Within the usual set theory of math (ZFC set theory), we can prove that we can assign every set a “cardinal number” that we call its cardinality. For finite sets we just assign natural numbers. For infinite sets we assign new numbers called alephs. We assign the natural numbers a cardinal that we call aleph_0.
These cardinal numbers come with an ordering relationship where one set has a cardinality larger than another set if and only if its associated cardinal number is larger than the other sets cardinal number. So, alepha_0 is larger than any finite cardinal, for example. There is a theorem called Cantor’s theorem that tells us we can continually produce larger and larger infinite cardinals in fact.
So, we know the reals have some cardinality, thus some associated cardinal number. We typically call this number the cardinality of the continuum. The typical symbol for this cardinality is a stylized (fraktur) c. Since aleph_0 is countable, every aleph after aleph_0 is uncountable. By definition aleph_1 is the smallest uncountable cardinal number. The continuum hypothesis just asks if aleph_1 and c are equal.
As an aside, it is provable that c has the same cardinality as the powerset of the naturals. We let the cardinality of the powerset of a set with cardinality x be written as 2^x. Then we can write the continuum hypothesis in terms of 2^{aleph_0} and aleph_1. The generalized continuum hypothesis just swaps out 0 and 1 for an arbitrary ordinal number alpha and its successor in this new notation.
Infinity cannot be divided, if it can then it becomes multiple finite objects.
It really depends on what you mean by infinity and division here. The ordinals admit some weaker forms of the division algorithm within ordinal arithmetic (in particular note the part about left division in the link). In fact, even the cardinals have a form of trivial division.
Additionally, infinite sets can often be divided into set theoretic unions of infinite sets fairly easily. For example, the integers (an infinite set) is the union of the set of all integers less than 0 with the set of all integers greater than or equal to 0 (both of these sets are of course infinite). Even in the reals you can divide an arbitrary interval (which is an infinite set in the cardinality sense) into two infinite sets. For example [0,1]=[0,1/2]U[1/2,1].
If infinity has a size, then it is a finite object.
Again, this is not really true with cardinals as cardinals are in some sense a way to assign sizes to sets.
If you mean in terms of senses of distances between points, in the previous link involving the extended reals, there is a section pointing out that the extended reals are metrizable, informally this means we can define a function (called a metric) that measures distances between points in the extended reals that works roughly as we’d expect (such a function is necessarily well defined if either one or both points are positive or negative infinity).
My degree is in math. I feel pretty confident in saying that you are tossing around a whole bunch of words without actually knowing what they mean in a mathematical context.
If you disagree, try the following:
What is a function? What is an injective function? What is a surjective function? What is a bijection?
In mathematics, what does it mean for a set to be finite?
In mathematics, what does it mean for a set to be infinite?
I’m willing to continue this conversation if you can explain to me in reasonably rigorous terms what those words mean. I’ll help you do it too. The link I sent you in my previous post that mentions cardinal numbers links you to a wikipedia page that links to articles explaining what finite and infinite sets are in the first paragraph.
To be clear here, your answer for 2 specifically should rely on your answer from 1 as the mathematical definition of a finite set is in terms of functions and bijections.
Here are some bonus questions for you to try also:
In mathematics, what does it mean for a set to be countable?
In mathematics, what does it mean for a set to be uncountable?
None of this includes the correct answers to the questions I asked you. I’m not going to read anything else from you until you correctly answer the questions I asked.
If there are infinite numbers, then there’s 3 in there somewhere.
No, this is not true. Just because you have infinitely many numbers in some collection, doesn’t mean one of the numbers in your collection has to be 3.
Look at the number line. There are infinitely many numbers on the number line between 1 and 2. For example 1+1/2, 1+1/4, 1+1/8, … are in there (among many others). But all of the numbers between 1 and 2 are strictly smaller than 3, so none of them can be 3.
Alternatively, there are infinitely many numbers strictly smaller than 3, none of which are 3 either.
If 3 is not there then it’s not infinite.
Well consider the set of numbers 3+1, 3+2, 3+3, 3+4, … (the set of integer numbers strictly larger than 3). This set of numbers is also infinite and does not contain 3. So a set being infinite doesn’t imply it must contain the number 3.
I considered reading and responding to this big long word salad you sent me, but I realized you were just further demonstrating the three points from my last post. Lmao, good luck.
Edit: Feel free to show me you learned the definitions I asked you about by answering my list of definition questions I posed to you a while ago by the way. I’m still fine with continuing if you do that.
I understand that you feel learning new things is hard. I sympathize with you. Lets start with a real easy one. High school algebra students often learn what mathematical functions are. You can handle that right? Tell me the mathematical definition of a function. Oh! Oops, I have accidentally linked you to a place where you can find the definition I’m asking you for in the first paragraph. Well, no going back now. Feel free to copy and paste the first paragraph of that link here.
Hmm, I wonder if there is a link between functions and finite/infinite sets? Oh gosh golly, perhaps they are related in some way? Almost like the definition of one requires some notion of the other?
My dear friend, I am very big fan of the back-pedaling you’re doing here. I want to also point a couple things out to you.
I’ve never argued that mathematics has a concept of finite or infinite numbers, or not. All that I have argued is that what the math world identifies as infinite, is not actually infinite when applied to the real world.
This is blatantly untrue. You can certainly play the post-hoc “oh but I meant…” game and slowly change your argument to be something different, but what you said originally is not what you are suddenly now claiming here and your lack of logical precision or clarity in the claims you make is certainly not my fault or my problem. Consider taking a course in mathematics to firm up your logical argumentation skills?
Let me remind you of a couple other claims you have made beyond what you are suddenly now pretending you claimed:
“Infinity cannot be divided, if it can then it becomes multiple finite objects.”
“If infinity has a size, then it is a finite object.”
“There is no infinityA or infinityB there is just infinity itself.”
“The statement ‘some infinities are bigger than other infinities’ is an illogical statement”.
“The mere statement that there are multiple infinities, negates either objects identification as being infinite, and reduces both objects to finite objects (more word salad follows)…”
Of course you have made a bunch of other claims in your weird psycho-babble word salad too. These are just some highlights.
Lets consider this thing you just said here though: “what the math world identifies as infinite, is not actually infinite when applied to the real world”. You know, this sounds very familiar. It is almost like my very first comment to you was “It really depends on what you mean by infinity and division here.” Real wild stuff huh? Almost like it is important to be clear on the definitions and senses of the words we are using right? Like we should be clear on what exact definitions we mean yeah? Hmm… This sounds so familiar.
As much as I’d love to make fun of you more while you rediscover arguments for/against mathematical platonism I’d rather move on.
As an engineer I deal with recursive functions, code that can run indefinitely. But as an engineer I understand that the code that is running needs an initiation point, the point at which the code is initially executed, and I understand that the seemingly infinite nature of the code, is bound to the lifespan of the process that execute it, for example, until the process is abruptly stopped, or power is taken away from the computer the process is running on. A lifespan invalidates the seemingly infinite nature of the code, from a practical sense. When you start to understand this, and then expand your focus to larger objects like the universe itself, you start to understand the finite nature of the material world we live in.
Loving the assumption here that I have no background in CS or software engineering.
I understand that mathematicians deal with abstraction. I deal with them too as an engineer. The difference is that as an engineer I have to implement those abstractions within the real world. When you do this enough times you will start to understand the stark differences between the limited hypothetical worlds math is reasoned about, and the very dynamic world the real world, that those math solutions are applied to. The rules of hypothetical worlds are severely limited in comparison to the real world. This is why it’s very important for me to define the real world boundaries that these math problems wil be applied to.
I don’t think claiming practical experience as an engineer as justification for misunderstanding and drawing faulty conclusions from basic mathematics is really the gotcha you think it is here. On the contrary, if you really do have a background in engineering, then you should know better and it is now my opinion that the people who have taught you mathematics and the basics of engineering have done you a serious disservice for not teaching you better. Misunderstanding mathematical models is textbook bad engineering. What you are doing here is using your engineering background to justify why it is okay for you to be a shitty engineer.
I’m used to working with folks, like yourself, that have a clearly hard time transitioning from a hypothetical world to the real world.
Who is having the trouble? I’m not the one stumbling over basic things that children learn in high school algebra like what the definition of a function is.
This is why I have respond with civility, and have looked past your responses insulting tone.
Oh yes, clearly my tone is insulting, but yours has never once been insulting. You pure beautiful angel you. If only the rest of us could be such a pure and sweet soul like you. I’ll be sure to only speak to you in the kindest and sweetest ways so that I don’t hurt your very precious and delicate feelings in the future.
I understand it’s a fear response of the ego, and I don’t judge you for it. I understand that it’s difficult to fight with the protection mechanisms of the ego.
I’m sorry kind and gentle prince, but I can’t help but point out that the projection here from you is very entertaining. I’m so very sorry for any hurt this may cause your poor delicate feelings.
Part of the goal here isn’t even mastery of the language itself. Exposure to new cultures is important. Being able to empathize with how hard learning a language is is also important.
This issue is not a black and white speak english vs not kind of thing. There’s no shortage of immigrants that speak english perfectly well sans a minor accent but are discriminated against and treated poorly anyway for not being native speakers.
Edit for those who downvoted: I am a native english speaker. I have been discriminated against based on my regional accent. Only a fucking fool would think the same things don’t happen to nonnative speakers.
If you have a fixed collection of processes to run on a single processor and unlimited time to schedule them in, you can always brute force all permutations of the processes and then pick whichever permutation maximizes and/or minimizes whatever property you like. The problem with this approach is that it has awful time complexity.
Edit: There’s probably other subtle issues that can arise, like I/O interrupts and other weird events fwiw.
eigenspaces (mander.xyz)
wholesomeness (lemmy.world)
That wasn't a microwave safe bowl... (lemmy.world)
I figured it was a 50/50 shot the bowl was microwave safe. We did not eat the beans…
[Solved] Algebraic Solutions to Graphical Trig Problems? (mander.xyz)
I’ve been knocking out the trig problems in this section with minimal difficulty so far, but I’ve run straight into a brick wall on this “Algebraic” part. I’m asked to find sin(x)=0 between [0,2π). If I graphed the unit circle this would be a trivial exercise to show sin(θ)=0 when θ=0 or π....
hmmmm (mander.xyz)
deleted_by_author
OnLy GoD cAn JuDgE mE (slrpnk.net)
That's not how graphs work (lemmy.world)
Credit
Htop too (lemm.ee)