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Topic: Failing Linear Algebra:
Replies: 91   Last Post: Jan 10, 2007 12:56 PM

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 Russell Blackadar Posts: 586 Registered: 12/12/04
Re: Failing Linear Algebra:
Posted: Apr 30, 2004 9:51 PM

On 01 May 2004 00:34:06 GMT, Anonymous wrote:

>Marc:
>

>>The first time, I spent in a linear algebra course (Kaiserslautern, 1986)
>>introduced sets, groups, rings and fields, modules and then vector spaces.
>>(in that order)

>
>Odd. We did sets and then groups in algebraic structures. We touched upon
>rings, I think, but I don't remember what they are. Have no idea about fields,
>if they're not vector fields, which someone already said they aren't.

If you can't give the definition of a field, then why do you think you
know what a vector space is? Look carefully again at the definition
of v.s. and see if there are any words in that definition whose
meaning you don't know.

(But probably for your purposes this semester it will suffice to know
a couple examples of fields, two or three of them to be specific,
instead of the definition. But no kidding, you *do* have to know at
least what I'm talking about here. "A vector space over __" -- fill
in the blank.)

Here's an exercise. Is the set C of complex numbers with the usual
complex addition a vector space over C? If so, find a basis. Is C a
vector space over R? If so, find a basis. What dimension is it? Is
the set of real numbers with normal addition a vector space over C?
(Answer is no, why?) Can you ever have a vector space over Q (the set
of rational numbers)?

Never
>heard of modules.
>
>We did vectors, then vector spaces, then subspaces. Then spans, dependence,
>bases, matrices, determinants, then eigenvalues and vectors. It's odd because
>Schaum's Outlines does it in a different order. And Cliff's does an even
>different order.

So? It's all stuff you have to learn. It all hangs together in a big
network -- the field isn't linear. (Puns intended.) Pick an
organization that works for you; that's why there are different books.

>
>>By the time we met vector spaces, the notion of image and kernel as well
>>as the proof of the first homomorphism theorem

>
>"Homomorphism theorem"? Is this what's called "Dimension Theorem" now?

No, it's something in group theory. Not entirely unrelated, though.

>
>>But nowadays nobody needs to be afraid of modules in a linear algebra course,
>>unless it is only a MMC (Matrix Manipulation Course).
>>

>
>What are modules? And why are they related to marix manipulation?

They aren't. Modules are a generalization of vector spaces, where
"field" is generalized to "ring". Every vector space is a module, but
not vice versa.

His point was that modules are not a difficult concept at all, and
anybody who can grasp the subject matter of linear algebra (in a
course like yours) should have no trouble understanding them. But in
a strictly MMC course there would be many students who would find them
scary. I don't know where that puts you personally; certainly right
now (since you don't remember what a ring is) you would be scared.
But fear not, looks like modules won't hit you this semester anyway.

BTW, I, for
>one, would love to have taken a linear algebra course that was all MMC. I love
>the stuff!

Yes, it can be fun. But once you know how to do it, it gets tedious
and then thank heavens, you can leave it to the computers. Hopefully
in time, you will find the abstractions even *more* fun, and anyhow
there is plenty of MMC you have to do in manipulating those
abstractions. Just read some of the postings here by the real
mathematicians, and you will see matrices galore.

Date Subject Author
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Thomas Nordhaus
4/24/04 Dave Rusin
4/25/04 Jonathan Miller
4/25/04 Felix Goldberg
4/24/04 Daniel Grubb
4/28/04 Tim Mellor
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 gersh@bialer.com
4/29/04 Daniel Grubb
4/29/04 Dave Rusin
4/28/04 Guest
4/29/04 Guest
4/28/04 Guest
1/10/07 David C. Ullrich
4/29/04 Dave Rusin
4/28/04 Guest
1/10/07 Law Hiu Chung
1/10/07 Dave Seaman
1/10/07 Marc Olschok
1/10/07 George Cox
4/28/04 Guest
1/10/07 Dave Rusin
4/28/04 Lee Rudolph
4/28/04 Guest
4/28/04 Guest
1/10/07 Marc Olschok
1/10/07 Toni Lassila
4/29/04 Guest
1/10/07 M L
1/10/07 Thomas Nordhaus
4/30/04 Guest
1/10/07 David C. Ullrich
1/10/07 Toni Lassila
4/30/04 Guest
1/10/07 George Cox
1/10/07 Marc Olschok
4/30/04 Guest
4/30/04 Guest
4/27/04 Guest
1/10/07 Thomas Nordhaus
1/10/07 David C. Ullrich
1/10/07 Dave Rusin
1/10/07 David C. Ullrich
5/9/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 Marc Olschok
5/10/04 David C. Ullrich
4/27/04 Guest
1/10/07 Thomas Nordhaus
4/27/04 Guest
1/10/07 magidin@math.berkeley.edu
1/10/07 David C. Ullrich
1/10/07 Marc Olschok
1/10/07 David C. Ullrich
1/10/07 Tim Mellor
4/28/04 Daniel Grubb
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 David C. Ullrich
4/28/04 Dave Rusin
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 Marc Olschok
4/24/04 Wayne Brown
4/24/04 Thomas Nordhaus
4/24/04 David Ames