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Topic: Failing Linear Algebra:
Replies: 54   Last Post: Jan 10, 2007 12:47 PM

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 Guest
Re: Failing Linear Algebra:
Posted: Apr 27, 2004 3:13 PM

rusin@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<c69sdb\$oi7\$1@news.math.niu.edu>...
> 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?

>
> I would bet that the single practice problem you need to work on is,
> "What is a vector space?"
>
> Our students "do well on the very first exam" because that's the part
> of the course where we warm up with techniques for solving linear
> systems of equations and such topics. But our LA course is also our
> students' first course in which abstractions, axioms, and proofs play
> a significant role. They often stumble because (among other problems)
> they don't realize they need to _memorize_ definitions _precisely_.
> So you can do a little self-assessment here to figure out whether
> what you're missing is bits of topics or the core idea: can you,
> right this minute, define what a vector space is?
>
> dave

Dave, this may be my problem. I did decent on the first exam because,
like you said, it was solving linear systems, echelon form, linear
dependences--easier stuff like that. I've always been a little
confused with the concepts, but I think I may finally be getting a
grip on what a vector space is:

It's a group of vectors that can be multiplied by any scalar and/or
added together in any way, and whatever possible combinations that can
result is the "vector space" for that group of vectors. This is how I
understand it. For vectors in R^2, a plane is formed ("spanned"???)
by the vector space. For vectors in R^3, a solid area is formed by
the vector space. It gets difficult for me to move into dimension 4.
While I understand that the same concepts hold, there's no more
physical picture I can use to visualize what's happening. Is my
understanding of "vector space" sufficient enough? Am I missing
anything?

I know I am still struggling with the concepts of "span" and "basis".
The weird thing is that I'm alright with the more advanced stuff;
matrices, determinants, eigenvalues, eigenvectors. I'm a little hazy
with diagonalization because it's the newest thing we've done. I know
it's got something to do with the eigenvalues of a special type of
matrix.

I guess I could also really use some help with understanding how a
mapping gets converted into a matrix, and then how to solve it. I
understand matrix multiplication and can do it well. But the concepts
of image, kernel, and isomorphism and how they relate to the
mappings/matrices seem to be lost on me. The odd thing is that I
fully understand the definitions of "kernel" and "image" as they were
applied in algebraic structures, but I don't get how they apply to
linear really. "Isomorphism" is a concept I never understood in
algebraic structures or linear algebra.

Thanks to the people who posted book suggestions, but I'm hesitant
Notes guide to linear algebra, a 2003-version of the Schaum's outline,
and I even have an old 1968 version of Schaum's that my grandmother
used when she majored in math. Cliff's has been helpful, but too
basic. Schaum's seems almost too advanced; it's great that they solve
all the problems, but sometimes the explanations are lacking. I find
that I do much better at math problems if I can first figure out how
to solve a certain type of problem and then go back and try to
understand the concepts behind it, rather than the other way around.
Schaum's examples don't allow for this, because they assume you've
problems.

Maybe, if it isn't too much to ask, would anyone here be willing to
post some problems relating to mappings/kernel/image/isomorphims
and/or eigenvalues/eigenvectors, and I can attempt to solve them with

Date Subject Author
4/22/04 Guest
4/22/04 Michael N. Christoff
1/10/07 Gerry Myerson
1/10/07 Jonathan Miller
1/10/07 Guest
1/10/07 David C. Ullrich
1/10/07 Acid Pooh
1/10/07 Guest
4/23/04 Brian Borchers
4/27/04 Guest
1/10/07 maky m.
4/26/04 David Ames
1/10/07 Guest
1/10/07 Michael Stemper
1/10/07 maky m.
4/23/04 Porker899
4/27/04 Guest
1/10/07 Abraham Buckingham
1/10/07 Mitch Harris
1/10/07 Guest
1/10/07 Grey Knight
1/10/07 Guest
1/10/07 Toni Lassila
1/10/07 Thomas Nordhaus
1/10/07 George Cox
4/28/04 Dave Rusin
4/28/04 George Cox
4/28/04 George Cox
4/29/04 Marc Olschok
4/29/04 Mitch Harris
4/29/04 Robert Israel
4/28/04 Guest
4/29/04 Guest
1/10/07 Dave Rusin