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

 Messages: [ Previous | Next ]
 Guest
Re: Failing Linear Algebra:
Posted: Apr 28, 2004 6:23 PM

Russell:

>>It's a group of vectors that can be multiplied by any scalar and/or

>
>Not just any way -- there are precise conditions that the sum must
>satisfy. Look for the axiomatic definition in your book(s). It's not
>at all difficult to memorize. Do it.

I already have memorized it. The space has to be closed under addition and
scalar mult, and contain the zero vector. Therefore, it satisfies those 8
properties:

*v* + *w* = *x* (in the space)
c*v* = c*v* (in the space)
0*v* = 0
1*v* = *v*

*v* + *w* = *w* + *v*
(*v* + *w*) + *x* = *v* + (*w* + *x*)
*v* - *v* = 0
c(s*v*) = (cs)(*v*)

>The algebra is the important thing, not the picture. And your current
>understanding of the algebra is insufficient to keep you from getting
>confused on the upcoming test.

But, I did so well in algebra in 7th-9th grades. How much different can this
be? I've got the definitions well enough: I got a B- on the first exam. It's
the *concepts* since image/kernel/basis that have confused me, like I said.

>>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 like the term "linear transformation" and I think you should use it
>too;

OK. See? I never knew what that term meant. Now I do.

>Do you know
>what restrictions I'm talking about? f(a+b) = f(a)+f(b) and
>f(ca) = cf(a) of course. That, by definition, is what makes the
>mapping *linear*.

Right. And also the 0 vector.

>Try working it out for the 90-degree rotation I mentioned
>above, and then try your matrix out on some 2D column vectors to see
>if they really do turn 90 degrees when you multiply by the matrix.

If something is rotated 90 degrees, the first point, cosine, goes from 1 to 0.
And sine goes from 0 to 1. I still don't really get how to represent that in a
matrix.

Theta(pi/2) of (0,1) becomes (1,0). Does that mean the matrix is just:

/ 1| 0|
\ /
2X1

?

>(I am assuming you multiply with matrix on the left and column vector
>on the right, as is done in Schaum's outline; hopefully that is how

Yes.

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