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Linear Algebra and Geometry: Memorization required?

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I''m in the midst of catching up on some linear algebra, thereafter heading into intensely studying geometry. And I''m a bit confused... I''ve re-noticed that there a LOT of Thereoms and formulas. My questions linger on these: How much to memorize? How much to just reference instead? Point is, I aim to be excellent at these math topics. I just do not quite know where to draw the line and say "Okay, I have remembered the pt. slope formula: (y -y1) = m(x - x1), now I will just go back to a book to look up the rest of the Thereoms and formulas." I am particularily interested in discoursing with anyone who has done well in these subjects, or math in general, to attain an outside perspective. Lastly, I realize that you must first UNDERSTAND how every works with the various formulas and thereoms before truly taking advantage of them. Again, my curiosity in regards to memorization comes after ascertaining the methodology behind the math. Advice, comments, quirky jokes are all welcome. ;-) A shortcut is the longest distance between two points.

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> How much to memorize?

Nothing. If your studying for an exam you need to learn things to recall them in an exam, but in the real world I find there''s little point.

Better to study a topic, e.g. by reading it, until you UNDERSTAND it. Then try using it, keeping to hand whatever references you need. You will quickly find that you start to remember the key points so you no longer have to look them up, while more obscure formulae/rules are never learned. It''s far easier to find out which things are worth learning this way than trying to learn them all by rote.

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I have a slightly different advice. You don''t have to learn anything (not much difference so far ). You should make yourself familiar enough with this subject that you can redo any formula or proof in a matter of seconds. I don''t even know the point-slope formula by heart! But there is no need.

The only non-trivial theorems (the ones that you may not be able to prove in seconds after you really understand the subject) are two:
- The rank of a matrix is the same as the rank of the it''s transposed.
- Jordan theorem.
And most people don''t use them on a daily basis.

The thing with linear algebra is that it is quite hard while you are studying it, and you only really get to understand it after two or three years of using it everywhere.

Work hard, and good luck.

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In linear algebra, do problems in it out on paper and your computer. Get a good grasp of the basics and don''t bother about the anal details. You can look them up when you need to.
For instance, you should know about what matrices do and the different operations commonly done with them(dot product, determinant, etc)
The common operations you should be able to do with only minimal lookup.

Thats my advice at least.

Bugle4d

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In going with what everyone else has said here, I believe that once you have attained a throrough understanding of the subject, you should be able to prove all of you need to know. Exceptions to this would be hard theorems that are used intermitantly through your study. The best satisfaction comes when the connectedness between plane geometry, linear algebra, and coordinate geometry can be found that will allow you to solve many of the problems that are discussed frequently on this forum. If you ever get satisfied, jump into number theory or something...

Brendan

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I would agree that working the problems is the best approach. It is like programming. Reading a book on programming and even memorizing all the syntax doesn''t mean you know how to program. Often what you read is not what they wrote. Starting out you have to work problems that test your understanding and over the long haul you have to use it on a regular basis or you will forget it.

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If your in on it for problem solving or programming or perhapes something similar you don''t need to remember anything on how to do something but you should always know that you can use them to do a certain task. of course this applies to some things more then others.

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My best Maths teachers have always said "Maths is all about practice". This means that if a question comes up in the exam which you''ve done a million and one times before you''re likely to remember it.

The brain is great at generalising things (and according to a book I''m reading that''s a fundamental part of how it works) so if you work out the slope of a billion lines you''ll be able to do a billion different ones. The same with pretty much everything e.g. integration (big one!), differentiation blah blah.

I would also agree that doing lots of problem solving improves your problem solving ability.

I personally don''t remember the equation for the slop of the line. I think in terms of the gradient is how much it goes up in a certain distance e.g. up/across or m = (y-y1)/(x-x1).
You''ll agree that y-y1 is a change in the up direction and x-x1 is the change in across direction And voila you get the gradient m. Most things are like this.

You''ll find as you get further through maths you learn more and more general formulae.

Oh BTW I disagree that there are only two non-trivial theorems. I have little experience with them but I''m pretty sure many of the axioms are far from trivial e.g. proving 1=1 etc.


-Meto

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I see this topic has became pretty popular, so I decided to drop my two cents in here as well.

Algebra, geometry, statistics, topography- beleive it or not all of these are quiet easy to understand. Once you understand the fundamental reasoning behind every division of math any formula that you might see (even for the first time) comes in naturally. Memorizing is pointless (from my pov) because it only makes you force the information in.

Instead of memorizing a theorem/proposition you should try to understand it. After that nothing seems as hard anymore, because it is only logical

Oh yeah- and the slope formula- that is one of the fundaments of linear euclidian geometry so you don''t even have to think about it, it pops up in your head naturally (at least for me).

EPHERE
www.ibmr.net

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quote:
Original post by Metorical

Oh BTW I disagree that there are only two non-trivial theorems. I have little experience with them but I''m pretty sure many of the axioms are far from trivial e.g. proving 1=1 etc.


-Meto


The whole point of axioms is that they can''t be proven... that''s why they are axioms. Although you can assume different axiom sets and prove one set from another. Godel showed that even with axioms, maths is inconsistent (i.e. there exist statements which can''t be proven or disproven; isn''t the Riemmann hypothesis one of those?).
But I do agree that not all the proofs are obvious... Cayley-Hamilton (every matrix satisfies it''s own characteristic equation) comes to mind...

"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley

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