Find transformation

1. Jun 15, 2011

Shackleford

I'm not exactly sure how to find the transformation. The professor wrote something different in class. I know [T]α is what you multiply with the "new" basis to get the transformation of the components of the "original" basis. In this case, it's simply still alpha.

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110615_211957.jpg [Broken]

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110615_204346.jpg [Broken]

Last edited by a moderator: May 5, 2017
2. Jun 15, 2011

lanedance

so guessing here, and abusing a little notation but hopefully it helps..

for a given matrix A you should able to write in the alpha basis:
$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = q\vec{\alpha}_1+pq\vec{\alpha}_2+rq\vec{\alpha}_3+sq\vec{\alpha}_3 = \begin{pmatrix} p \\ q \\ r \\ s \end{pmatrix}_{\alpha}$$

then apply the T transform which is already written in the alpha basis

3. Jun 15, 2011

lanedance

to further understand the alpha basis, note that you could consider A expressed in the standard basis, call it s, and write
$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} +b\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} +c\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} +d\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix}_s$$

4. Jun 15, 2011

lanedance

updated above

5. Jun 15, 2011

Shackleford

Oh, I see what you're doing.

6. Jun 15, 2011

lanedance

what's not the alpha basis?

you need to solve for q,p,r,s which give the components in the alpha basis

7. Jun 15, 2011