# Homework Help: 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

### Shackleford

The components are already given.

8. Jun 16, 2011

### lanedance

the way i read it (open to interp):
- the components of A in the standard basis are given
- the components of the operator T in the alpha basis is given

so i think you need to express A in the alpha basis, or express T in the standard basis

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