David C. Ullrich <firstname.lastname@example.org> wrote in message news:<email@example.com>... > On 22 Apr 2004 15:44:12 -0700, Anonymous wrote: > > >I'm currently a math major and am taking linear algebra, but I'm in > >serious danger of failing. I just don't get it! Is this newsgroup a > >place to come to ask questions and get information about learning > >math? Or is there somewhere more appropriate to go? I've always had > >trouble with vectors, and I think I fell apart sort of right at the > >beginning of linear algebra (although, I did manage to get a B- on the > >very first exam). I've got another exam next week. What can I do? I > >don't get all the terms, concepts, and jargon. Anyone know how to > >make learning linear algebra easier and more practical? Anyone got > >any practice problems?
Thanks for the post. I've read it all the way through and agree and understand what you're saying. About the definitions, I like your definition for "basis": "independent spanning set" because it's short and simple. Therefore, it's *easy* to memorize. My text book would tend to defend the basis in a really, really abstract way like: "Let S [contained in symbol] V be a subset of V. If *x* [element symbol] S = a1x1 + a2x2 + ... + a^nx^n = *0*, then *x* is a basis if a^i (1 < i < n) = 0 for all a^i and Span(V) = S.", or something really wordy and convoluted like that. I mean, I eventually understand what the definition is saying. But, "independent spanning set" is just so much easier, IMO.
> > My answer to this is a little long. I hope you'll read all of it: I've > taught that course probably 20 times in the last 29 years; > I know two things from experience: (i) what I have to say is > good advice (ii) you're not going to believe it's good advice. > The long part of this post is an attempt to explain why it's > good advice (also some actual empircal evidence that it > _is_ good advice, whether you believe it or not). > > The advice is what Rusin already said: next time you take the > course _learn_ the definitions, _precisely_. Word for word. > You need to know the definitions well enough that when you > see the word "basis" the phrase "independent spanning set" > pops into your head _immediately_, without a moment's > thought. > > Probably it's too late for that this semester, because there's > a _large_ number of definitions you need to learn. You > need to learn each one, _precisely_, as soon as it comes > up in class. > > Why: > > I teach that class a lot. A lot of students have a lot of trouble. > I always give the class the advice above. Almost all the > students simply ignore the advice - if they do bother to > try to actually learn the definitions it's just the day before > the test. I think this is because "memorize" is sort of a > dirty word - our attitude these days is that we're not > supposed to be memorizing things, we're suppposed to > be "understanding" them. I tell me students they need > to learn the definitions and they say "you mean _memorize_ > them?", as though they can't believe I'd suggest such > a thing. > > One way to look at it is this: you're trying to learn a > new _language_. When you take German you don't > find anything strange about the fact that you have to > simply memorize what the words mean - _after_ you > memorize what the words mean you can understand > sentences written in German; trying to understand > German prose _before_ simply learning what the > words mean is simply ridiculous. But students try > to do the equivalent thing in linear algebra. > > The thing is you're dealing with an _abstract_ > subject. If you're studying potatoes you don't > need a definition of "potato", because you already > know what a potato is. But if you're studying > vector spaces, linear operators, etc, you do need > to know the definitions, because a vector space > is not something you can point to, a vector space > is exactly what the definition says it is. > > Of course _understanding_ the definitions is the > actual goal. But in an _abstract_ subject what it > _means_ to understand a definition is to see > exactly how it fits in with the _other_ definitions. > If you know all the definitions, _precisely_, then > it may happen at some point you _will_ see how > they all fit together, and at that point voila, you > have understood the definitions! But if you don't > know _exactly_ what the definitions say then it > _can't_ happen that at some point you will see > exactly how they all fit together, because your > fuzzy versions of the definitions will _not_ fit > together the way they're supposed to! > > So much for the abstract explanation. Now for > the empirical evidence: > > I teach that course a lot. I always ask for a lot > of definitions on quizzes and tests. Not because > I think that being able to recite the definitions should > actually be the goal, but because I know that knowing > the definitions is essential to being able to work the > problems, and I also know that the students don't > believe that. (What really puzzles me is that they > don't find the fact that I'm going to be asking for > the defintions on quizzes and tests to be enough > motivation to learn them - it happens that I ask > for the definiton on "basis" four weeks in a row, > all they have to say is "independent spanning > set", and some students will get the very same > question wrong four weeks in a row. Whatever...) > > Now, not all the people here who teach this class > do ask for definitions on quizzes and tests. Here's > the good part: It's happened several times that > someone takes the course from me and flunks. > He retakes the course from someone else. > Even though the other professor _doesn't_ > ask for definitions on quizzes and tests he > decides to give it a try. I run into the student > later, and he tells me that he tried "my" way, > and sure enough it worked! He re-took the > course, _learned_ all the definitions rock-solid, > and he got an A or a B in the course! > > NOTE that he didn't get that A or B the second > time because he got the definitions right on > tests. The other professor wasn't asking for > definitions. He got the A or B because sure enough, > it turned out that knowing the definitions allowed > him to actually do the problems. > > Try it next time. (Or don't, but it's the only way, > honest).
There won't be a next time, unfortunately. If I fail it now, I have no way of fitting in the four more upper-level maths that I need for the major, since linear algebra is a prerequisite for them all. If I fail now, I'd have to drop the math major. You guys be the judge. Everything I've typed here (all definitions, including the sample one above in this post) is from my own mind. I haven't consulted the book for anything. Is my knowledge of the concepts good enough? I have a test on Friday and then a final exam in two weeks. Together, they're worth about half of my grade. If I ace them both, I can bring my borderline D/F grade up to at least a C, especially when the curve is considered. I'm not the only one doing poorly in the class. On the second exam, 1/2 the class got above an 80, and the other 1/2 got between a 43 and 72.
So, there was a retake exam. Basically everyone who got below a 72 took it. BTW, I got a 46 on the original (so, 2nd lowest in the class, probably) and a 51 with the retake included (a 53 on the retake itself, but the original 46 was weighted in too, somehow). I got a 79 on the first exam, so my test average is, what, a 65-ish? I'm sure my homework doesn't help or hurt that. So, like I said, if I get around 90's on the last two tests, my final grade would be around 77/78, right? Is it all just a lost cause now? The question is, will I be able to ace the last two exams?