Too easy of an undergrad?

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dantetheinferno
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Too easy of an undergrad?

Postby dantetheinferno » Thu Jun 02, 2011 9:15 am UTC

Hi there,

I'm currently a Junior at my university, and I was looking around at some of the topics in this mathematics subforum and realized for quite a few of the questions, I had absolutely no idea how to solve them. My university isn't well known for it's mathematics program (more known for its engineering or health care programs). I was wondering if maybe I just haven't taken the correct level classes yet. I've so far taken Calc 1-3 (I saw people saying they learned limit episolon definitions in calc 1, and proved almost all of the theories and I was wondering how the hell they would be able to do that in an introductory calculus class.), Diff EQ, Complex Analysis, Linear Algebra 1, and Abstract algebra along with one other proof based class. For instance, in my linear algebra class, we never mentioned isomorphisms unlike in this thread:

http://forums.xkcd.com/viewtopic.php?f=17&t=71107

Am I just being paranoid, or should I be able to solve this topic by now?

Token
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Re: Too easy of an undergrad?

Postby Token » Thu Jun 02, 2011 2:26 pm UTC

Should you be able to answer the question? I'd guess not - I don't know exactly how your class system works, but it's a level of linear algebra that's above what I'd call "basic". I'm surprised you don't know what an isomorphism is, though...
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nehpest
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Re: Too easy of an undergrad?

Postby nehpest » Thu Jun 02, 2011 2:57 pm UTC

My intro linear algebra course never mentioned isomorphisms either, though to be fair it was 5 hours a week to cover linear algebra and differential equations and was more computational than proof-based.

If I could offer up my own experience, I've actually learned a fair bit of math just by reading this forum and Googling the topics I didn't know about or understand. There are a lot of resources online for undergrad math, including a wealth of practice problems to work through - this has helped me more than any one course I've taken.
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z4lis
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Re: Too easy of an undergrad?

Postby z4lis » Thu Jun 02, 2011 3:01 pm UTC

I'm not surprised that your first linear algebra class didn't equip you with the theory needed to solve that problem. Mine certainly didn't. However, I am surprised that you took an abstract algebra class that was proof-based and didn't learn what an isomorphism is there. Hmmm. What did you cover?
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

gorcee
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Re: Too easy of an undergrad?

Postby gorcee » Thu Jun 02, 2011 3:14 pm UTC

A good curriculum won't teach you everything, but it will give you the tools to be able to learn anything.

Chances are, if your school is more engineering-based, your math classes are more applied math-based. Which is fine. If you want to understand the more theoretical stuff, you will have to ratchet up your formalism a little bit, but you already have the foundation of knowledge in place.

Also, don't feel bad regarding epsilon-delta stuff. It's a relatively recent, and controversial, movement to get away from teaching e-d proofs in intro Calc courses.

Tirian
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Re: Too easy of an undergrad?

Postby Tirian » Thu Jun 02, 2011 3:36 pm UTC

Linear Algebra is a very mysterious subject. For some treatments (like mine), it was an abstract course, but I've read more than enough books that just talk about how to manipulate R^3 enough to be able to do physics and engineering problems that I've come to decide that maybe I'm the outlier. Then if you make it to the second course, you'll see that R^k is ALSO a linear space, no matter which natural number you use for k! :shock:

To give a specific example, I wouldn't be surprised if there were graduates of linear algebra who knew linear isomorphisms only in the form of the process of switching between bases and not as an invertible mapping between linear spaces that preserves the linear structure.

The nice thing about this is that whichever course you took, you can always find an inexpensive book or website that describes the theory from the other perspective which can be read through very quickly to give you a comprehensive overview. I think that's a good idea even if you learned it "the right way" the first time. :P

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z4lis
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Re: Too easy of an undergrad?

Postby z4lis » Thu Jun 02, 2011 3:37 pm UTC

gorcee wrote:A good curriculum won't teach you everything, but it will give you the tools to be able to learn anything.


This sums up my viewpoint on my own education. Even with the best professors, the only thing you will ever learn is what you teach yourself. Everything else is simply there to make the process easier.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Yakk
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Re: Too easy of an undergrad?

Postby Yakk » Thu Jun 02, 2011 6:25 pm UTC

There is enough material in "linear algebra", "calculus" and the like that you can do entire courses without touching more than a small amount of theory. You could also jump into the theory whole-hog.

As engineers and physicists need the "applied" parts of linear algebra and calculus, most places end up having a "generic" course that doesn't cover much if any theory, and just exposes students to the applied stuff. If you lack a large enough "pure" or "theoretical" mathematics program, they simply don't have the students to have a theoretical branch of the courses.

You'll note that the person posting the linear algebra course was tutoring it -- and it was probably a theoretical branch -- so it isn't surprising if you don't have the tools. :)

The way you do epsilon-delta definitions in a calc 1 class is that you don't teach much computation, you start with students who are all self-motivated to learn mathematics, and deal with students who where all near top of their class in high school. Doing this to a bunch of people who want to be engineers, accountants or chemists is possibly pointless and overly painful to them.

Now, note that some of the quirkier "less applied" parts of linear algebra ends up showing up in the strangest places (I've seen engineering courses where you take the characteristic polynomial of the transition-flow matrix of a flow graph to solve "practical" problems...), so it isn't completely pointless.

Also note that you can easily have a grad course called "linear algebra", and blow the minds of grad students with the content. The name of a course just describes the type of subjects, it doesn't define the course.
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LLCoolDave
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Re: Too easy of an undergrad?

Postby LLCoolDave » Thu Jun 02, 2011 7:08 pm UTC

As the topic linked in the first post was by me I felt the need to chime in to give some more context to the problem. First of all, as I stated there, the question wasn't what was intended to be on the exercise sheet and the answer isn't at all obvious as none of us tutors and in fact even the professor teaching the course wasn't certain about the answer once we noticed that the question was about an isomorphism and not just any endomorphism. Both the solutions presented in the topic aren't at all elementary. The explicit one requires a clever idea to express the eigenvectors in the given form as well as some knowledge on roots of unity, which are usually covered at the end of a course on field extensions or in number theory. The other approach uses the Jordan canonical form of a complex endomorphism, which is a fairly deep result in introductory linear algebra courses. Both of them require knowledge on the special structure of the complex numbers.

Just to give you a heads up, none of the 93 students turned in a correct solution to the exercise, most of them made very wrong conclusions in their approaches. I do not think there is a solution to the problem they could be expected to find with their current level of knowledge. We're also teaching a fairly abstract courrse in linear algebra which is taken by pure math students, mathematical economy students and physics and comp sci students that are interested in a stronger mathematical education. All other students have more applied math courses to take.

I also have to repeat the statement that any math education is just going to give you some threads of interesting topics to follow, the more you learn the more you will figure out how much you still don't have a clue about. In a sense I feel I know less now than I did two years ago, and I quite frankly don't see how you can aquire the necessary knowledge to reach the frontline of research in just a single lifetime.

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Re: Too easy of an undergrad?

Postby Nat » Thu Jun 02, 2011 7:54 pm UTC

we do the epsilon-delta definitions in out first-semester calc (of course, we have an incredibly abstract university, and there's separate courses for physics students). Still, it always sounds weird when people tell me about places that teach nonformal calc to pure math students. It sounds more like high school then uni, and (for pure math students) it is neither particularly interesting nor particularly useful.

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nehpest
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Re: Too easy of an undergrad?

Postby nehpest » Thu Jun 02, 2011 8:01 pm UTC

For what it's worth, my high school AP Calc AB class started with, and spent 2-3 weeks on, e-d proofs.
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pizzazz
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Re: Too easy of an undergrad?

Postby pizzazz » Thu Jun 02, 2011 11:33 pm UTC

Yeah, it sounds to me like you just have a particularly applied math curriculum, without much pure theory. On the other end of the spectrum, my first year calc class touched, in varying degrees of detail, on set theory, topology, fields, and now vector spaces (we defined isomorphisms today). We spent less than half the year on "typical" calculus topics (limits, derivatives, integrals, and sequences/series).

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Re: Too easy of an undergrad?

Postby achan1058 » Fri Jun 03, 2011 12:08 am UTC

pizzazz wrote:Yeah, it sounds to me like you just have a particularly applied math curriculum, without much pure theory. On the other end of the spectrum, my first year calc class touched, in varying degrees of detail, on set theory, topology, fields, and now vector spaces (we defined isomorphisms today). We spent less than half the year on "typical" calculus topics (limits, derivatives, integrals, and sequences/series).
Actually, to me his curriculum seems normal, and it's yours that is particularly theoretical. Most school's lower level math courses are designed for the majority of science students, instead of for math students, especially since it isn't always practical to separate the students given class sizes and all that. I presume that this isn't the case in your school, and the courses for math majors are drastically different than that for the rest of the students that needs to take math courses.

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Re: Too easy of an undergrad?

Postby pizzazz » Fri Jun 03, 2011 12:27 am UTC

achan1058 wrote:
pizzazz wrote:Yeah, it sounds to me like you just have a particularly applied math curriculum, without much pure theory. On the other end of the spectrum, my first year calc class touched, in varying degrees of detail, on set theory, topology, fields, and now vector spaces (we defined isomorphisms today). We spent less than half the year on "typical" calculus topics (limits, derivatives, integrals, and sequences/series).
Actually, to me his curriculum seems normal, and it's yours that is particularly theoretical. Most school's lower level math courses are designed for the majority of science students, instead of for math students, especially since it isn't always practical to separate the students given class sizes and all that. I presume that this isn't the case in your school, and the courses for math majors are drastically different than that for the rest of the students that needs to take math courses.


I did say my class was at one end of the spectrum. And while the OP's class is probably more towards normal than most, I'm a bit surprised that a linear algebra course doesn't even define isomorphisms.

Although my class is even more theoretical than most of the introductory calc sequences, this is still a pretty theory-heavy school. If you want to graduate, you will have to do e-d proofs or similar, and if your major requires math beyond the core requirements (ie anything more than introductory calculus), you will have to take a proof-based course. I would definitely say it is not the case that there are low-level math classes designed for science majors. On the other hand, there are plenty of people who take the more theoretical stuff despite it being unnecessary.

dantetheinferno
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Re: Too easy of an undergrad?

Postby dantetheinferno » Fri Jun 03, 2011 10:06 pm UTC

For my abstract algebra class, other sections of the class do cover isomorphisms/ homomorphisms, our professor just chose not to.

Reading through this thread makes me realize that my program is a lot more oriented towards the applications based part of Mathematics. Most of my fellow students are in secondary education rather than solely mathematics. In my calculus classes, you're expected to have memorized all of the important taylor series, derivatives, etc. and be able to use them very, very quickly.

Thanks for the reassurance everyone. :)

dissonant
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Re: Too easy of an undergrad?

Postby dissonant » Sat Jun 04, 2011 4:19 am UTC

At my university, first-year math courses seem to be mostly for the benefit of engineering students rather than mathematicians.

The net result of this is that calculus is treated quite informally, and many of the motivating examples are applied rather than pure. ("Calculate the work done by a particle moving through a vector field, etc.") I actually find this approach quite beneficial as it is good to study calc. in its natural habitat. The downside to all of this however, is that qualities such as logical rigour, which are necessary to mathematicians but not so for engineers, are sacrificed. C'est la vie.

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Nic
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Re: Too easy of an undergrad?

Postby Nic » Sun Jun 05, 2011 2:57 am UTC

dantetheinferno wrote:For my abstract algebra class, other sections of the class do cover isomorphisms/ homomorphisms, our professor just chose not to.

Okay, now I would be worried. Just out of curiosity, what did you do in that class? Write out multiplication tables? Trying to do algebra without being able to determine if two groups are isomorphic is like learning precalc without knowing what an equals sign means.

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agelessdrifter
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Re: Too easy of an undergrad?

Postby agelessdrifter » Mon Jun 06, 2011 12:16 am UTC

I'm coming out of two years at community college and just about to begin my Junior year studying mathematics (and physics), and I had the same concerns as you, Dante. One thing I decided early on was that I would at least read every section of the textbook for whatever math courses I took, regardless of whether the professor chose to cover them. I just recently finished taking Linear Algebra, and we didn't cover eigen values/vectors really, beyond the definition and how to find them for small matrices -- we also didn't talk about isomorphisms or changes base (in detail), or how to diagonalize matrices, as well as quite a bit of other material, but sure enough, that stuff's all in the (fairly scant) text book. I'm not sure how much of that non-mandatory material I could ace a test on right now, but I feel more comfortable with the prospect of encountering it again in the future knowing that I've been exposed to it and worked on it a bit.

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Re: Too easy of an undergrad?

Postby crates » Mon Jun 06, 2011 5:51 pm UTC

If you can find the answers, is that not good enough? I think we place too much emphasis on memorization in general.

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mdyrud
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Re: Too easy of an undergrad?

Postby mdyrud » Mon Jun 06, 2011 5:54 pm UTC

There's a big difference between finding the answers and knowing why the answer is what it is. I've spent the last two years unofficially tutoring a fellow who was in a math class he didn't belong in. Given a problem that had been addressed in class or the textbook, he could do it, but because he didn't understand the mechanisms behind finding the answer he couldn't solve any unfamiliar problems. Does that mean you need to be able to prove every single theorem you encounter in a calculus course? Of course not. But there does need to be some knowledge of the theory behind the solutions.

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Re: Too easy of an undergrad?

Postby Jyrki » Tue Jun 07, 2011 5:33 am UTC

dantetheinferno wrote: For instance, in my linear algebra class, we never mentioned isomorphisms unlike in this thread:

viewtopic.php?f=17&t=71107

Am I just being paranoid, or should I be able to solve this topic by now?


As a poster in that thread let me give a few words, too. The solution based on Jordan canonical form is something that a linear algebra student might come up with provided that in addition to JCF they have met the exercise: Prove that for all natural numbers [imath]n[/imath]
[imath]\left(\begin{array}{cc}1&x\\0&1\end{array}\right)^n=\left(\begin{array}{cc}1&nx\\0&1\end{array}\right)[/imath]
as well as its consequence
[imath]\left(\begin{array}{cc}\lambda&1\\0&\lambda\end{array}\right)^n=\left(\begin{array}{cc}\lambda^n&n\lambda^{n-1}\\0&\lambda^n\end{array}\right).[/imath]
In a typical linear algebra course JCF may be mentioned at the end of the chapter on eigenvalues and diagonalization. It is usually not studied in detail, because its proof properly belongs to a more advanced course. I don't think it would be fair to expect the students of a first course in linear algebra to use it in a proof like this. An exposure to the phenomena standing in the way of our ability to diagonalize a matrix (and to the extent that JCF remedies the situation) is needed, but not covered in a freshman course.

The second solution needs an ingredient from representation theory of groups. A cyclic group suffices here, and many might recognize it as an application of discrete Fourier analysis. As a teacher of a first course in linear algebra I would not dream of putting it up as an exercise except in the following simpler form with my usual dose of embedded hints:

=====

Assume that [imath]f:C^n\rightarrow C^n[/imath] is a linear mapping and that a vector [imath]x\in C^n[/imath] is an eigenvector of the mapping [imath]f^2[/imath] belonging to the eigenvalue 1. Show that

A) If [imath]y=x+f(x)\neq0,[/imath] then [imath]y[/imath] is an eigenvector of [imath]f[/imath] belonging to the eigenvalue 1.

B) If [imath]z=-x+f(x)\neq0,[/imath] then [imath]z[/imath] is an eigenvector of [imath]f[/imath] belonging to the eigenvalue -1.

C) The vector [imath]x[/imath] can be written as a linear combination of eigenvectors of [imath]f[/imath].

=====

After working thru a problem like this, there might some hope for solving the general case. Still, the roots of unity are needed, preferrably in a context, where DFT is used, which is not (at least not here) covered in a freshman course.

Anyway, to reiterate, this stuff is beyond a first course in linear algebra. I assume that it ended up on a problem sheet accidentally. Even with easier variants like the problem above leading up to it, if a freshman can solve it, he/she is something special (or has been exposed to a lot of stuff in advance).


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