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

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 Guest
Re: Failing Linear Algebra:
Posted: Apr 29, 2004 4:15 PM

Russell:

>Think basis vectors. What is the basis we are using here?
>

>>
>>Theta(pi/2) of (0,1) becomes (1,0).

>
>I think you've rotated in the wrong direction;

Oops. I didn't rotate in the wrong direction. I had a total brain fart and
forgot which point comes first. Right...what I meant to say was that (1,0)
rotates to (0,1).

>The vector (0,1) transforms to (-1,0);

Right. Stupid mistake.

Still, I don't get how this result is written in matrix form. How would the
mapping theta (pi/2) be represented in matrix and basis form?

>Since this y-vector was our second basis vector, we
>now have the *second* column of our matrix. But we still need to find
>the first column.

OK, so this, I guess:

1 0
0 1
-1 0
0 -1

But, the second two are just multiples of the first two. So, they're
unnecessary in a basis, right? So, the basis would just be: {(1,0), (0,1)} for
a 90 degree rotation?

Question: if the rotation was by (pi/4) instead, would the basis have to be:

Another area where I'm having some problems is with projection mappings and
matrices.

>Hopefully you have
>worked the matrix out in your head by now, so I'll write it down:
>
> 0 -1
> 1 0
>
>That is, the first column is a unit in the y direction, and the second
>column is a unit in the -x direction.

Why would the first x-value be negative though?

>Let's see if this matrix
>works for some arbitrary vector, say, (2,1). Express that vector as a
>column vector, multiply by our matrix, and you get the answer (-1,2)
>expressed as a column vector, right?

0 -1
1 0

times (2, 1) =

(0 -1 , 2 - 0) = (-1, 2). Got it!

So this 90 degree rotation basis has to work for ANY unit vector then, no
matter the direction it points in?

>Plot the two points (2,1) and
>(-1,2) on some graph paper and draw in the lines from the origin; what
>angle do you see?

Yup, I can visualize that. They're 90 degrees apart. Wait...something else
just hit me. (-1, 2) isn't a unit vector. How come our 90 degree rotation
basis works anyways? Would it work for a vector of any length?

>In fact
>one of the things you may be asked to do is, given a matrix in one
>basis, compute the matrix for the same linear transformation in a
>different basis. I won't go into that here; it's in all of your books
>and hopefully you have enough insight now to be able to understand

I think I just learned how to do that yesterday. You convert the matrix into
the identity matrix, and whatever has happened to the original identity matrix
as a result of the conversions equals the inverse of the original matrix.
I.E., you convert A into A^(-1) using I. You can check that A*A^(-1) = I, and
then you know you're right.

With A^(-1), you multiply one side of the standard basis by A^(-1) and the
other side by A, and that gives you basis alpha in terms of basis beta. Right?
I'm a little hazy here, I think. But do I have the geneal idea down?

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