# Compute the change of basis matrix

• Mar 19th 2011, 06:10 AM
posix_memalign
Compute the change of basis matrix
I wish to compute the change of basis for the following:

B = {$\displaystyle \begin{bmatrix} 1\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ -1 \end{bmatrix}$}

C = {$\displaystyle \begin{bmatrix} 0\\ 1 \end{bmatrix},\begin{bmatrix} 2\\ 1 \end{bmatrix}$}

How can I compute the change of basis matrix $\displaystyle P_{C \leftarrow B}$?

There are supposedly (at least) two ways to do this.

I know how to do this by expressing the vectors in B in terms of the basis C; when I do that I get the result that:

$\displaystyle P_{C \leftarrow B}$ = $\displaystyle \begin{bmatrix} 1/2 & -3/2\\ 1/2 & 1/2 \end{bmatrix}$

Is this correct at all? Although even if it is correct I'd like to be able to do it using a different approach I got described.

Supposedly I first have to transform the B coordinates to the canonical coordinates and thereafter the canonical coordinates to C coordinates, this should give $\displaystyle P_{C \leftarrow B}$ as $\displaystyle P_C^{-1}P_B$
Could someone show me how to do this for this example?
• Mar 19th 2011, 09:29 AM
tonio
Quote:

Originally Posted by posix_memalign
I wish to compute the change of basis for the following:

B = {$\displaystyle \begin{bmatrix} 1\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ -1 \end{bmatrix}$}

C = {$\displaystyle \begin{bmatrix} 0\\ 1 \end{bmatrix},\begin{bmatrix} 2\\ 1 \end{bmatrix}$}

How can I compute the change of basis matrix $\displaystyle P_{C \leftarrow B}$?

There are supposedly (at least) two ways to do this.

I know how to do this by expressing the vectors in B in terms of the basis C; when I do that I get the result that:

$\displaystyle P_{C \leftarrow B}$ = $\displaystyle \begin{bmatrix} 1/2 & -3/2\\ 1/2 & 1/2 \end{bmatrix}$

Is this correct at all? Although even if it is correct I'd like to be able to do it using a different approach I got described.

It is correct and hopefully you understand what you did here: you express each element of the new

basis as a lin. combination of the new basis and the matrix we're looking for is the transpose of the

coefficient matrix obtained in the first step.

This is, as far as I can tell, the easiest and quickest method.

Supposedly I first have to transform the B coordinates to the canonical coordinates and thereafter the canonical coordinates to C coordinates, this should give $\displaystyle P_{C \leftarrow B}$ as $\displaystyle P_C^{-1}P_B$
Could someone show me how to do this for this example?

As I can see this is exactly the same as before: to express the basis B in terms of the canonical

coordinates is just to take the matrix B formed with the elements of B, and then...etc. Exactly the same!

Tonio
• Mar 19th 2011, 10:49 AM
posix_memalign
Quote:

Originally Posted by tonio
As I can see this is exactly the same as before: to express the basis B in terms of the canonical

coordinates is just to take the matrix B formed with the elements of B, and then...etc. Exactly the same!

Tonio