mangoe@umcp-cs.UUCP (Charley Wingate) (03/05/86)
A side note on math and CS-ers: I too have noticed that computer science types tend to have trouble with math. Here at UMCP they are required to take the first two calculus courses, and you've never heard such wailing and gnashing of teeth (except in the elementary CS courses, which baffle EVERYONE). I'm not sure why this is, but I suspect it has to do with the supposed effects of woodpeckers on buildings built by CS people. C. Wingate
agw@garfield.columbia.edu (Art Werschulz) (03/05/86)
In article <77@umcp-cs.UUCP>, mangoe@umcp-cs.UUCP (Charley Wingate) writes: > A side note on math and CS-ers: > > I too have noticed that computer science types tend to have trouble with > math. Here at UMCP they are required to take the first two calculus > courses, and you've never heard such wailing and gnashing of teeth (except > in the elementary CS courses, which baffle EVERYONE). I'm not sure why this > is, but I suspect it has to do with the supposed effects of woodpeckers on > buildings built by CS people. > > C. Wingate I too have notced this. I don't think this is a local phenomenon, having taught at both University of Maryland Baltimore County and at Fordham University. I'm still looking for a good explanation. Art Werschulz ARPAnet: werschulz@cs.columbia.edu USEnet: ... seismo!columbia!lexington!agw BITNET: werschulz%cs.columbia.edu@wiscvm ATTnet: Columbia University (212) 280-3610 280-2736 Fordham University (212) 841-5323 841-5396-- Art Werschulz ARPAnet: werschulz@cs.columbia.edu USEnet: ... seismo!columbia!lexington!agw BITNET: werschulz%cs.columbia.edu@wiscvm ATTnet: Columbia University (212) 280-3610 280-2736 Fordham University (212) 841-5323 841-5396
gds@mit-eddie.MIT.EDU (Greg Skinner) (03/06/86)
Actually, I have found that some hackers have trouble with math also.
Particularly calculus. I took the first year of freshman calculus in
my first semester, partially because I had a head start in high school
but mostly so I could get it out of the way and get on with my CS
courses :-) Anyhow, I have always wondered why some hackers had
trouble with calculus. As far as I could tell, I was using the same
kind of logic in solving a calculus problem as in a programming
problem. However, I knew of some hackers that dropped out of school
because they couldn't get through calculus, even though they were able
to sort and hash lists, search trees, etc.
Actually, this brings me to a deeper question. I have always wondered
why some hackers just didn't grit their teeth and suffer through the N
years to get their degrees. Considering the fact that it was easy for
them to grind out their programs, they could have coasted through all
their programming projects and devoted some of the energy that went
into their programs into learning linear algebra, modern algebra, and
all that other stuff. I can remember putting in a lot of time on all
my courses -- I didn't like putting in all that time so much but I
considered it necessary, and felt so much better for really
understanding it. (Yet, I suppose to someone whose goals are only to
put out portable, fast code, without understanding why it is portable
and fast, understanding is of little importance.)
I don't know, maybe I am missing some fundamental issue regarding the
difference between those who sweated out all the courses towards their
degree and those who didn't. I would appreciate some enlightenment
though. When I was an undergrad, I would have loved it if I could
whip up fast, portable code for my assignments. I envied those who
could. I was just curious why it didn't carry over into their other
courses.
--
It's like a jungle sometimes, it makes me wonder how I keep from goin' under.
Greg Skinner (gregbo)
{decvax!genrad, allegra, gatech, ihnp4}!mit-eddie!gds
gds@eddie.mit.educrm@duke.UUCP (Charlie Martin) (03/07/86)
As far as Computer Science people having more trouble with Math -- Knuth had a real interesting article on algorithmic thinking vs. mathematical thingking (and if I were in my office I could give a citation, but I'm not.) The basic point of the article was that there are two styles of thhought involved, and that being good at one *does not* imply that one is good at the other. As one who is good at algorithmic thinking and had to struggle with math, I can certainly attest to it. But this does suggest that the curricula for CS may be a little misplaced: perhaps they are trying to teach horse cavalry drills to helicopter cavalry? -- Charlie Martin (...mcnc!duke!crm)
jma@duke.UUCP (Jon M. Allingham) (03/07/86)
In article <77@umcp-cs.UUCP> mangoe@umcp-cs.UUCP (Charley Wingate) writes: >A side note on math and CS-ers: > >I too have noticed that computer science types tend to have trouble with >math. Here at UMCP they are required to take the first two calculus >courses, and you've never heard such wailing and gnashing of teeth (except >in the elementary CS courses, which baffle EVERYONE). I'm not sure why this >is, but I suspect it has to do with the supposed effects of woodpeckers on >buildings built by CS people. > >C. Wingate That's interesting. I went to an engineering school and was required to take the 2 year calculus sequence (with the exception of the very last quarter which was a second course in differential equations) that everyone took, plus discrete math and a course in probabiltiy and statistics. We also had to take 1 year of physics and 2 quarters of chemistry or biology! As a graduate student now, there are times when I wish I had had even more math, in particular the differential equations course which we didn't have to take! I don't see how a school can teach Computer Science without requiring a hefty dose of math. You can't do much analysis of algorithms, communtications systems etc. without a fair understanding of probability and statistics, and really understanding P & S ( as opposed to looking up formulae in books ) requires a good understanding of integral and differential calculus + infinite series etc. It's amazing even at the graduate level how many people can't handle double integrals and functions with anything other than 'x' as the variable! In case you're wondering, my degree was a B.S. in Information and Computer Science (Ga. Tech), and the School of ICS was in the College of Science and Liberal Studies, not the College of Engineering. (Lucky for us, otherwise we would have had to take 'def. bods' and a whole bunch of other engineering courses!) -- Jon M. Allingham Duke University/AT&T Bell Laboratories {mcnc,decvax}!duke!jma
laura@hoptoad.uucp (Laura Creighton) (03/07/86)
In article <1194@mit-eddie.MIT.EDU> gds@mit-eddie.MIT.EDU (Greg Skinner) writes: > >Actually, this brings me to a deeper question. I have always wondered >why some hackers just didn't grit their teeth and suffer through the N >years to get their degrees. Considering the fact that it was easy for >them to grind out their programs, they could have coasted through all >their programming projects and devoted some of the energy that went >into their programs into learning linear algebra, modern algebra, and >all that other stuff. I can remember putting in a lot of time on all >my courses -- I didn't like putting in all that time so much but I >considered it necessary, and felt so much better for really >understanding it. (Yet, I suppose to someone whose goals are only to >put out portable, fast code, without understanding why it is portable >and fast, understanding is of little importance.) > >I don't know, maybe I am missing some fundamental issue regarding the >difference between those who sweated out all the courses towards their >degree and those who didn't. I would appreciate some enlightenment >though. When I was an undergrad, I would have loved it if I could >whip up fast, portable code for my assignments. I envied those who >could. I was just curious why it didn't carry over into their other >courses. You have missed it totally, Greg. Computer Programming and Mathematics both require thinking, yes. But not the same sort of thinking. Case in point -- linear algebra. I am in the unfortunate position that I can't get my csc degree without the course in Linear Algebra -- and I could knock it off in one semester if I could get that course. But I can't. I have taken it three times, dropped it once, and failed it twice. The last time with a 49%. (All I need is a pass. One more mark would have done it, but I couldn't get it for blood nor money). Now I am willing to believe that there is a way to approach linear algebra which lets me use my intellect to understand the course and thus work out problems ``from general principles''. But *nobody* has been able to teach me this. I spent one semester taking no other courses but Linear Algebra (I still was working, but I picked a time when I could coast along at work) so tyhat I could finally get rid of this obstacle. That is the year that I got 49%. And the reason why is that, without understanding (which I was never able to get, despite working my ass off) linear algebra is nothing but sheer memorization. Now check out your hackers -- they can't memorise without understanding. I know more than a dozen people who don't have their degree because they couldn't get past linear algebra. I don't know *anybody* who turns out fast, portable code who doesn't understand why it is fast and portable. I just see absolutely no connection between this understanding and linear algebra. If there is one, and you can get it across to me, I might be willing to give linear algebra another crack -- after going to all that work, I would like to have the csc degree. But until then, I've had it. I just can't memorise that much. By the way, it is not that I don't have the mathematical background I have the ``calculus courses for physics majors'' and logic courses galore. Something which may be related -- when I solved a problem in any branch of mathematics until I hit linear algebra there was this tremendous rush ``aha!'' and a warm rosy glow. Since I, by and large, live for those experiences, I found programming and cs theory attractive -- I can the same feeling out of getting my code to work, or understanding of a real interesting theory or algorithm. (To tell you the truth, I have Hofstadter's malady, and can get the same feeling by contemplating recursion). But not for an nanosecond did I ever get this feeling after solving a linear algebra problem. Indeed, I didn't even get a much of a sense that it was done. (The thing I liken it to is to reducing a very large polynomial fraction. You factor the numerator and denominator and cancel out the common factors and are left with a reduced expression. As a kid I used to do them all afternoon for the joy of it. The elegence! The simplicity! But I can't get that feeling out of linear algebra. On the other hand, I can really get it out of taking a mess of code and making it protable and then speeding it up. The simplicity! The functionality! Wow.. and my mind starts glowing again...) What I don't understand is people who have never felt this at all. I have explained ``why mathematics is beautiful'' to countless people and got no reaction at all. It is all so much ``linear algebra'' to them I hope that there are other areas of their lives which give them the same feeling, though -- to live life without ever feeling this ``aha!'' feeling sounds very, very bleak to me. -- Laura Creighton ihnp4!hoptoad!laura utzoo!hoptoad!laura sun!hoptoad!laura toad@lll-crg.arpa
doug@terak.UUCP (Doug Pardee) (03/08/86)
> Actually, this brings me to a deeper question. I have always wondered > why some hackers just didn't grit their teeth and suffer through the N > years to get their degrees. Considering the fact that it was easy for > them to grind out their programs, they could have coasted through all > their programming projects and devoted some of the energy that went > into their programs into learning linear algebra, modern algebra, and > all that other stuff... > > I don't know, maybe I am missing some fundamental issue regarding the > difference between those who sweated out all the courses towards their > degree and those who didn't. I would appreciate some enlightenment > though. A fair question. Since I was a hacker who never finished my degree, I'm at least slightly qualified to suggest some answers. First, that all-knowing philosopher "/usr/games/fortune" tells us that "Life is what happens to you while you're making other plans." In my case, a full-time job offer came along. Since I was one of five kids being supported by my mother on her GS-3 salary [everyone say "Awwww!"], an opportunity to switch from being a financial negative to a positive was hard to pass up, especially since my starting salary was more than my mother was making. And two years later, I got married, thereby closing the door on any chance of returning to starving studenthood. Second, after three years of college I was still a freshman. This was because the courses I needed to take were always closed by the time they were offered to freshmen. In retrospect, I can see that this was simply because the college I attended was *very* popular for junior college transferees. But at the time, I tried to fight the system, and lost regularly. I never had more than a 1/2-time workload. But more to the point, perhaps... I managed over the course of five years of "part-time" study to take all of the math/science/computer courses I needed for my degree (Math w/ CS minor). And *still* I was barely a sophomore. Ahead were three full-time years worth of Physical Education, History of World Religions, and other such courses that I had no interest in whatever. Continuing part-time, it would consume most of my free time for at least a decade. Not very appealing to a newly-wed. So, since my career was doing just fine without a degree, I simply dropped the matter. I haven't regretted it. For me, it was the right decision. -- Doug Pardee -- CalComp -- {hardy,savax,seismo,decvax,ihnp4}!terak!doug
g-rh@cca.UUCP (Richard Harter) (03/09/86)
In article <> gds@mit-eddie.MIT.EDU (Greg Skinner) writes: >...... >Actually, this brings me to a deeper question. I have always wondered >why some hackers just didn't grit their teeth and suffer through the N >years to get their degrees..... A note on this. I have learned, from painful experience, that there is a class of people who have a block about completing major projects. They may be brilliant and capable of very good work, but when it comes to that final push to get the job done, they can't do it, somehow. It is typical of people in this group that they don't complete college. One of the values of a college degree of any kind to an employer is that it is tangible evidence that the person in question can, in fact, carry a major task through to completion. Richard Harter, SMDS Inc.
flackc@stolaf.UUCP (Chap Flack) (03/09/86)
Laura Creighton writes: > in point -- linear algebra. I am in the unfortunate position that I > can't get my csc degree without the course in Linear Algebra -- and > I could knock it off in one semester if I could get that course. But > : > By the way, it is not that I don't have the mathematical background > I have the ``calculus courses for physics majors'' and logic courses > galore. Something which may be related -- when I solved a problem > in any branch of mathematics until I hit linear algebra there was > this tremendous rush ``aha!'' and a warm rosy glow. Since I, by and > large, live for those experiences, I found programming and cs theory > attractive -- I can the same feeling out of getting my code to work, > or understanding of a real interesting theory or algorithm. (To tell > you the truth, I have Hofstadter's malady, and can get the same > feeling by contemplating recursion). But not for an nanosecond did I > ever get this feeling after solving a linear algebra problem. Indeed, > I didn't even get a much of a sense that it was done. > > (The thing I liken it to is to reducing a very large polynomial fraction. > You factor the numerator and denominator and cancel out the common > factors and are left with a reduced expression. As a kid I used to > do them all afternoon for the joy of it. The elegence! The simplicity! > But I can't get that feeling out of linear algebra. On the other hand, > I can really get it out of taking a mess of code and making it protable > and then speeding it up. The simplicity! The functionality! Wow.. > and my mind starts glowing again...) I *hated* linear. I passed by the skin of my teeth and the mercy of my prof. I spent the term doing all sorts of mechanical procedures without ever feeling like I was *doing* anything. My roommate (a physics major, but I'm not trying to generalize) had no sympathy--when he had taken linear, he just learned the mechanics and went through them mechanically with no problem. Perhaps it's not unrelated that his programming gives the same impression: he has little sense of, or interest in, a beautiful algorithm.... Anyway, I tried not to think about linear algebra for a year after I had taken it. But then, as my term project in data structures, I was playing with some graphics programming (the Warnock hidden-line alg). I had to go back to my linear book and dig out all the stuff I'd learned and see that I could use it to really *do* things! Elegant things! Aha! And a great big rosy glow. Wow.... So I guess if you ever feel like taking it again, maybe you'd have more fun if you found some secondary sources on your own that you could read at the same time and see what neat things you can *do* with it. And the best of luck to you! -- --------------------- Chap Flack ihnp4!stolaf!agnes!flackc Carleton College ihnp4!stolaf!flackc Northfield, MN 55057
cs3031bm@unmg.UUCP ( slime face) (03/10/86)
In article <> gds@mit-eddie.MIT.EDU (Greg Skinner) writes: > ...I have always wondered why some hackers had >trouble with calculus. As far as I could tell, I was using the same >kind of logic in solving a calculus problem as in a programming >problem. However, I knew of some hackers that dropped out of school >because they couldn't get through calculus, even though they were able >to sort and hash lists, search trees, etc. I would consider myself a hacker (I have never started an assignment more than 3 days before it was due, with the vast majority started the night before, and about a 95% success rate.) and fortunately math has always been easy for me. I think the difference between calculus and all previous math classes is that calculus is not "obvious." I never did any homework in math (well, actually I did 23% of my calculus homework.) and got 'A's until calculus when I got a 'C'. I got so used to not having to work that I stopped working. I believe this is typical for most hackers. We believe that we're so smart that some nice person will give us a job and pay us $$$. I have seen a few of my friends turn into bums and decided that wasn't for me. >Actually, this brings me to a deeper question. I have always wondered >why some hackers just didn't grit their teeth and suffer through the N >years to get their degrees. Considering the fact that it was easy for >them to grind out their programs, they could have coasted through all >their programming projects and devoted some of the energy that went >into their programs into learning linear algebra, modern algebra, and >all that other stuff. I can remember putting in a lot of time on all >my courses -- I didn't like putting in all that time so much but I >considered it necessary, and felt so much better for really >understanding it. (Yet, I suppose to someone whose goals are only to >put out portable, fast code, without understanding why it is portable >and fast, understanding is of little importance.) I am suffering through the N years now. I am doing this to learn "obvious" things I haven't thought of yet, to find out what obvious things I know which are wrong, and to make sure I get a job doing interesting programming. I think that most of us care only about understanding, and the only reason we would care about portability is because our boss tells us to. Speed on the other hand I would consider understanding. I don't want to write a slow program, even if it loses lots of readability. We don't apply ourselves to other subjects because they don't interest us. I can bluff my way through most humanities, but most hackers can not. They feel that the extra $ they would make aren't worth the effort of doing something they don't enjoy, and I agree with them to an extent. I, unlike most, feel that the time I put in now won't neccessarily give me a better paying job, but it will give me a better chance of finding a more enjoyable job which makes college worth the time. This is unfortunate since most of my hacker friends work in fields totally unrelated to computer science, or don't work at all. A lot of good talent is going to waste because of our present school system, but no better alternative is obvious so it looks like they just get screwed by the system. Mr. Bill ______________________________________________________________ guess what this is: ZZ :wq quit quit! bye stop ^[ ? help ^Y ^Z ^C : :^[ exit eun w q q! ^D Arg!!!! __________________________________________________________ someone trying to get out of a silly editor
ebh@cord.UUCP (Ed Horch) (03/10/86)
>> [Greg Skinner] >> Actually, this brings me to a deeper question. I have always wondered >> why some hackers just didn't grit their teeth and suffer through the N >> years to get their degrees. Considering the fact that it was easy for >> them to grind out their programs, they could have coasted through all >> their programming projects ... > [Doug Pardee] >Continuing part-time, it would consume most of >my free time for at least a decade. > >So, since my career was doing just fine without a degree, I simply >dropped the matter. I haven't regretted it. For me, it was the right >decision. First, a reply to Greg: Being able to "coast" does not change the size of the hill. What I mean is that no matter how good you are, programming still takes time. I'm not talking about the hundred-line things you can knock off in two hours. I'm talking about large, multi-week, multi-deadline projects. For example, in the school I went to, their second-quarter assembly class was considered the "weed-out" course. The second half of the course consisted of writing a relocating assembler for the Univac 1100 in Univac 1100 assembly. About 15% of the students managed to get an absolute assembler working, but only *one* student ever got the relocating assembler to run. One student in over ten years. This is where Doug comes in: I too had to wrestle with the school vs. work conflict. I ultimately quit school altogether. The reason was simple: After a sixteen-hour day, who has the energy to go home and do calculus? Certainly not I. The typical scenario went like this. A new quarter/semester is coming up, and things are going pretty smoothly at work, so I decide to take two or three classes. Three weeks into it, a crunch hits at work, and seventy-hour weeks are imminent. I immediately drop one course, while I can still withdraw-pass. Things get rough. An occasional ninety- hour week is thrown in with the seventy-hour weeks, so I try to drop another course. But now it's late enough in the quarter that I can only get a withdraw-fail. Things at work back off to only sixty hours a week, so I think I can pull out the one remaining class (in which, for obvious reasons, I'm quite far behind). BUT WHAT ABOUT THAT DEADLINE THE NIGHT BEFORE THE FINAL??? I take the final after having been up for two days, blow it, (even though I know the material) combine that with two programming assignments I never turned in, and presto! Yet Another D Or F, and my grade-point drops again. Now, before you think that I've blown it off for good, a quick update: I now work at a forty hour per week project, and it looks like it will stay that way. By fall I hope to be back at school. After all, maybe my father will then think I've risen above bumhood. Also there's an economic motivation: as a consultant, I have to carry professional liability (read malpractice) insurance. Without a degree, the premiums are absolutely extortionate. So, in tax writeoffs and reduced premiums I should recoup all expenses. But I'm digressing. To reiterate: Greg, you should understand that, whether for class, client, or employer, real, nontrivial software design and development is a time consuming process, only made worse by deadlines, and in many cases that leaves neither the time nor the emotional energy for hard study. In these cases, school must either be put off until conditions allow, or dispensed with completely. Them's the breaks. -Ed Horch {cord,bentley}!ebh
greg@harvard.UUCP (Greg) (03/10/86)
In article <6988@duke.UUCP> crm@duke.UUCP (Charlie Martin) writes: >The basic point of [Knuth's] article was that there are two styles of >thhought involved, and that being good at one *does not* imply >that one is good at the other. As one who is good at algorithmic >thinking and had to struggle with math, I can certainly attest to it. I for one don't like making a strict dichotomy between math and computer science. There is no one "way of thinking" in mathematics; there are approaches to each field of mathematics, and each approach requires different thought processes in varying degrees. The same is true of computers. Certainly some fields of mathematics are very algorithmic, like complexity theory, while some fields in computer science are very mathematical, like mathematical computer graphics. Some of you out there seem to think that the abilities of a coworker, colleague, or prospective employee can be reduced to a simple "University of Texas, 1982, B.S., C.S.". Well, a diploma by itself doesn't say anything about how much creative insight, self-discipline, ability to learn, common sense, "deep understanding", ability to work with others, experience, or broad knowledge its owner has. Whatever his or her diploma says, a college graduate may or may not have major side interests, which may or may not be appropriate for the particular position (s)he has or is applying for. For example, if you had a position open for a researcher in the state-of- the-art computer graphics, and Richard Feynman and my friend-Teddy-with-a- C.S.-degree applied for the job, which should you hire? My answer is Richard Feynman, because he is a genius, while Teddy is not, despite the fact that Feynman's degree is in physics. -- gregregreg
g-rh@cca.UUCP (Richard Harter) (03/11/86)
In article <> flackc@stolaf.UUCP (Chap Flack) writes: >Laura Creighton writes: >> in point -- linear algebra. I am in the unfortunate position that I >> can't get my csc degree without the course in Linear Algebra -- and >> I could knock it off in one semester if I could get that course. But >> : > >I *hated* linear. I passed by the skin of my teeth and the mercy of >my prof. I spent the term doing all sorts of mechanical procedures >without ever feeling like I was *doing* anything. > This whole interchange puzzles me. What are they teaching as linear algebra these days that makes it so hard? I would suppose that a linear algebra course would cover such things as vector spaces, linear transformations, matrix representation of linear transformations, and the principal properties of matrices (i.e., rank, diagonalization, eigenvalue-eigenvector representation, etc.). It might cover linear transformation groups also, which I admit can get hairy. Now I well understand that one can be blocked against a subject or find it intrinsically uncomfortable, but I find it hard to understand how linear algebra could be one of those subjects. Upon reflection, I will concede that it could be very dry if it were taught by one of those masters of aridity that infest university math departments. But it is so incredibly fundamental to applied mathematics that the concept of a linear system ought to become intuitive by extended exposure. Color me confused. Richard Harter, SMDS Inc.
jxs7451@ritcv.UUCP (03/11/86)
The funny thing about CS student having problems with math is that most math
students have big problems with comp sci courses.
jeff "in sunny downtown Rochester"(or something like that)
UUCP: {allegra,seismo}!rochester!ritcv!jxs7451
CSNET: jxs7451%rit@csnet-relay.ARPA
BITNET: JMS7451@RITVAXClaura@hoptoad.uucp (Laura Creighton) (03/12/86)
Linear Algebra -- boy is there a lot of difference in how the courses are
presented in different places...
I got a lot of mail about this subject -- and I am still gettting it.
Thanks for the mail, people. But from it I can see that what a lot
of people took in linear algebra was matrices, and calculating determinents,
and Eigenvalues and so on. Where I come from, matrtic manipulation and
determinents are taught in high school. And I learned about Eigenvalues
in physics courses. And I never had any problem with them -- indeed, one
of the most enjoyable things I have done in recent times is read the
microcode which went into an array processor which did this sort of thing.
(i found 2 bugs, too!).
But I had no idea that this had anythign to do with linear algebra. I got
a copy of Schaum's Outline for linear algebra, though and so I can see
that it must be..but this wasn't the course
What we got was pure theory. There were no numbers, except as subscripts
and superscripts. There were definitional questions, such as:
What is a vector space?
What is a sub-space? bases and dimensions? fields?
And a literal hundred lemmas and theories -- which is what I couldn't
memorize.
I found an example in some old papers I have. On the right hand side of
the page are notes to myself on how to convert a pdp-11 running PWB to
v7. And on the left side...
Let V be a vector space and suppose T and U are linear operators
on V such that
(a) U is onto
(b) The Null spaces of T and U are finite-dimensional.
Then the null space of TU is finite-dimensional and
dim(N(TU)) = dim(N(T)) + dim(N(U)).
proof: let p = dim(N(T)), q = dim(N(U)), and {u1,u2,...,up} and {v1,v2,...vq}
be bases for N(T) and N(U) respectively. SInce U is onto, we can
choose for each i (i =1,...,p) an element wi (- V such that
U(wi) = ui. Thus we obtain a set of p elements {w1,w2...,wp}.
Note that for any i and j, wi != vj, for otherwise
ui=U(wi)=U(vj)=0-a contradiction.
[I wrote it that way. I think now that it should be
...=U(vj)=0 -a contradiction!]
Hence the set
B={w1,w2,...wp,vi....,vq} contains p+q distinct elements.
To prove the lemma it suffices to show that B is a basis for
N(TU).
-------------
It goes on and on in that vein for another 40 lines.
I had page after page after page of this stuff to memorize. And you know
something? I still don't know what a vector space *is*... It is still
all so many words.
I think that if people want to discuss the beauty of linear algebra with
me we should do this in mail or in net.math. But I am curious -- is
this what other people had in linear algebra? Or number crunching?
--
Laura Creighton
ihnp4!hoptoad!laura utzoo!hoptoad!laura sun!hoptoad!laura
toad@lll-crg.arpalamy@utai.UUCP (Jean-Francois Lamy) (03/13/86)
I personally would say that the "very algorithmic parts of mathematics" are in
fact just as much part of Computer Science. As an aside, I would prefer using
the French word "Informatique", defined as "the science of logical and
automatic treatment of information". This has the benefit of not using the
word "computer", and to indicate that some kind of formal description of the
processes involved has to be possible.
I think that what distinguishes informatics as a field is a concern for the
specification of algorithms.
This includes the specification of the properties of the data they operate on,
the specification of the expected results, the specification of the
operational behaviour, as well as considering devices (abstract or physical)
capable of carrying out the execution.
--
Jean-Francois Lamy
Department of Computer Science, University of Toronto,
Departement d'informatique et de recherche operationnelle, U. de Montreal.
CSNet: lamy@toronto UUCP: {ihnp4,utzoo,decwrl,uw-beaver}!utcsri!utai!lamy
EAN: lamy@iro.udem.cdn ARPA: lamy%toronto@csnet-relayed@mtxinu.UUCP (Ed Gould) (03/13/86)
In article <609@hoptoad.uucp> laura@hoptoad.UUCP (Laura Creighton) writes: >Linear Algebra -- boy is there a lot of difference in how the courses are >presented in different places... > >What we got was pure theory. There were no numbers, except as subscripts >and superscripts. There were definitional questions, such as: > > What is a vector space? > What is a sub-space? bases and dimensions? fields? When I took the course that covered that material, it was called either "modern algebra" or "abstract algebra", I don't remember which. I happened to have a good professor, so I actually learned something (or I thought I did - I don't remember any of it now). I can easily see being confused by a less-than-perfect description of the ideas. When I took linear algebra, it was all matrices, determinants, and simultaneous equations, etc. Easy stuff. -- Ed Gould mt Xinu, 2910 Seventh St., Berkeley, CA 94710 USA {ucbvax,decvax}!mtxinu!ed +1 415 644 0146 "A man of quality is not threatened by a woman of equality."
gds@mit-eddie.MIT.EDU (Greg Skinner) (03/14/86)
As to the obviousness of calculus -- I can see one area which would
give some people trouble -- limits. If calculus was to be more to you
than just a mass of formulas and exercises (like doing determinants)
than you must have some appreciation for the concept of a limit. I
never had the chance to ask some of my hacker friends about their
experiences with limits -- after all, the concept is not algorithmic
in nature. But then again, most of P & S is not either, and I have
seen hackers come out of P & S with more success than calculus. I
don't recall having any trouble with limits -- I believed the pictures
of the delta-y/delta-x graphs, the rate of change physics problems,
and the Riemann integral pictures. After you get through all that it
is just a matter of setting up the integrals or derivatives properly
for the problem. As I stop to think abut it though, this is also not
algorithmic in nature, since it involves some spatial reasoning.
In one of my previous postings, I alluded to the fact that hackers
ought to be able to grind out class assignments quickly. I did not
mean to say that. What I meant was more along the lines of "since
you're applying those techniques to your pet projects anyway, why not
do it just for some class?" But you are right -- it does take time and
energy. I recall that in one of my courses "software enginering", up
to 40% of the grade was on testing and documentation. Usually it took
over 50% of my time to come up with a set of tests for my projects,
defend why the tests were sufficient to "prove" the correctness of my
code, and document all that stuff. One good thing came out of all
that -- I did become disciplined in how to track down bugs or other
potential problems, and how to code to check for unusual situations.
I would have rathered work on side projects than do all that testing
and documentation but I don't think it was all for naught.
One more thing -- the areas of linear algebra that deal with vector
spaces, rank, nullity, etc., I do not consider that "alegbra" in the
same sense that I consider group theory, field theory, isomorphisms,
etc. to be algebra. I much preferred tha "algebra" in linear algebra.
I don't suppose those other portions of linear algebra could be
considered geometry?
--
It's like a jungle sometimes, it makes me wonder how I keep from goin' under.
Greg Skinner (gregbo)
{decvax!genrad, allegra, gatech, ihnp4}!mit-eddie!gds
gds@eddie.mit.educrm@duke.UUCP (Charlie Martin) (03/14/86)
In article <767@harvard.UUCP> greg@harvard.UUCP (Greg) writes: >In article <6988@duke.UUCP> crm@duke.UUCP (Charlie Martin) writes: >>The basic point of [Knuth's] article was that there are two styles of >>thought involved, and that being good at one *does not* imply >>that one is good at the other. As one who is good at algorithmic >>thinking and had to struggle with math, I can certainly attest to it. > >I for one don't like making a strict dichotomy between math and computer >science. There is no one "way of thinking" in mathematics; there are >approaches to each field of mathematics, and each approach requires different >thought processes in varying degrees. As far as there being more that one way of thinking in mathematics -- I'm not sure I believe you. Have you any evidence? Don't take this tquestion too seriously, as I am also not certain that "ways of thinking" are well-defined.... >.... >Certainly some fields of mathematics are very algorithmic, like complexity >theory, while some fields in computer science are very mathematical, like >mathematical computer graphics. There certainly is no question that certain fields of math *can be understood algorithmically* -- that is certainly the way I understood integration and differentiation when I first learned about them. (Now I just use infinitestimals when I can -- shows you what getting interested in logic can do.) But *my point* is that there are two styles of thought, and some people come more easily to one than the other. I come more easily to algorithmic thinking, and less so to mathematical. Math is harder for me. But I still find it useful. >For example, if you had a position open for a researcher in the state-of- >the-art computer graphics, and Richard Feynman and my friend-Teddy-with-a- >C.S.-degree applied for the job, which should you hire? My answer is Richard >Feynman, because he is a genius, while Teddy is not, despite the fact that >Feynman's degree is in physics. >-- >gregregreg I'd hire Dick Feynmann because he's an artist, and thus has knowledge of the problem domain. But I understand he's pretty happy at CalTech, I don't think we could get him. -- Charlie Martin (...mcnc!duke!crm)
ladkin@kestrel.ARPA (Peter Ladkin) (03/15/86)
In article <9430@ritcv.UUCP>, jxs7451@ritcv.UUCP writes: > The funny thing about CS student having problems with math is that most math > students have big problems with comp sci courses. Hardly. Only the ones that aren't interested. I don't recall any math students having problems with analysis of algorithms, compiler design, theory of computation, logic or graph theory. And most don't have problems with the concurrency puzzles that constitute a significant fragment of operating systems courses. Where did you get this idea from? Peter Ladkin
ait@ingres.berkeley.edu.ARPA (Allen Tuan) (03/17/86)
In article <5826@kestrel.ARPA> ladkin@kestrel.ARPA (Peter Ladkin) writes: >In article <9430@ritcv.UUCP>, jxs7451@ritcv.UUCP writes: >> The funny thing about CS student having problems with math is that most math >> students have big problems with comp sci courses. >Hardly. Only the ones that aren't interested. Beaver: Gee Wally, I guess you could say that most computer science students don't do well at math because they aren't interested in math, huh? Wally: Hardly. Gee Beaver, don't you know anything? Math students are God's gift to the Earth, and if they are not good at anything, it is only due to lack of interest, and not, God forbid, due to lack of ability. Most computer science students, on the other hand, are scuzzes of the Earth because they aren't good at math (which as we all know, is infinitely more interesting than, say, programming). >I don't recall any >math students having problems with analysis of algorithms, >compiler design, theory of computation, logic or graph theory. >And most don't have problems with the concurrency puzzles that >constitute a significant fragment of operating systems courses. >Where did you get this idea from? > >Peter Ladkin Gee Pete, has it ever occurred to you that perhaps math students who have ventured into the realms of analysis of algorithms, compiler design, theory of computation, graph theory, operating systems, etc etc etc are precisely those who are *interested* in and have a *talent* for computer science. I too, if my memory happened to be as conveniently crippled as yours, can recall only computer science students who have interest and talent in math, and have had no problems in upper division math courses offered here at Berkeley. Enough sarcasm for now (sarcasm is one of my reactions to condescension on the part of alleged know-it-alls (yes, yes, I know, no more name calling)). My point is, sure, there are math students who do well in compiler construction and operating systems courses, just as there are computer science students who do well in abstract algebra, number theory, and the likes. But to conclude from that that (that that???) ALL math students are good at compiler construction or concurrent programming seems a bit foolhardy. After all, if a math student didn't do well in introduction computer science courses (if they *elected* to take those courses in the first place), he/she isn't likely to venture far into the computer science curriculum. Computer science students, on the other hand, are *required* to take some math courses, and since the love of math isn't what led them to computer science in the first place (at leat not in my case), it would seem reasonable that some (or perhaps a lot, depending on the school, I'd guess) computer science students would flounder in math. There, I finally got my two bits in. Allen [no .signature - flames to ait@ingres.berkeley.edu] on the other hand, are *required* to
jxs7451@ritcv.UUCP (03/18/86)
In article <5826@kestrel.ARPA> ladkin@kestrel.ARPA (Peter Ladkin) writes: >In article <9430@ritcv.UUCP>, jxs7451@ritcv.UUCP writes: >> The funny thing about CS student having problems with math is that most math >> students have big problems with comp sci courses. > >Hardly. Only the ones that aren't interested. I don't recall any >math students having problems with analysis of algorithms, >compiler design, theory of computation, logic or graph theory. >And most don't have problems with the concurrency puzzles that >constitute a significant fragment of operating systems courses. >Where did you get this idea from? > >Peter Ladkin sorry Peter, what you say might be true in general, but not at RIT. I am in fact a u-grad in a Comp math program here. Personally I do good in the CS courses, but many of my friends in the various majors here are not so good. I can think of 4 or 5 people off hand that have switched majors because of the CS courses they were required to take. And I know many more that are not good with it, but they suffer through it anyway. Then again "the ones that aren't interested" in something rarely do well at it. Which is probably part of the CS majors problem with math. jeff "in sunny downtown Rochester"(or something like that) UUCP: {allegra,seismo}!rochester!ritcv!jxs7451