# Change of Basis

• Dec 2nd 2010, 07:07 AM
BrianMath
Change of Basis
Let $\displaystyle V \subset R^3$ be the plane defined by the equation $\displaystyle 2x_1 + x_2 =0. Let T:R^3 \rightarrow R^3$ be the linear transformation defined by reflecting across V. Find the standard matrix for T.

This should be done using change of basis formula. The standard matrix for T, A, should be $\displaystyle A=PBP^{-1}$, where B is the matrix of the transformation with respect to some basis. I chose my basis to be two vectors that are a basis of V and a vector that is in V perp.

How do I find B and solve this problem?
• Dec 2nd 2010, 07:33 AM
Opalg
Quote:

Originally Posted by BrianMath
Let $\displaystyle V \subset R^3$ be the plane defined by the equation $\displaystyle 2x_1 + x_2 =0. Let T:R^3 \rightarrow R^3$ be the linear transformation defined by reflecting across V. Find the standard matrix for T.

This should be done using change of basis formula. The standard matrix for T, A, should be $\displaystyle A=PBP^{-1}$, where B is the matrix of the transformation with respect to some basis. I chose my basis to be two vectors that are a basis of V and a vector that is in V perp.

How do I find B and solve this problem?

You have made a good choice of the basis. The operation of reflection across V will leave vectors in V unchanged, and it will take a vector in V perp to its negative. So the matrix B should be $\displaystyle \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}$.
• Dec 2nd 2010, 07:45 AM
BrianMath
I tried doing this myself where B is $\displaystyle \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}$ and P is $\displaystyle \begin{bmatrix}1&0&2\\ -2&0&1\\ 0&1&0\end{bmatrix}$ where the first two columns of P are a basis of the plane V, and the third column is in V perp, however, my answer did not match the book's answer. Is my choice of P wrong?
• Dec 2nd 2010, 07:53 AM
BrianMath
oh wait, nevermind!! I was just looking at the wrong solution at the back of the book. HAHA my bad. But this means I understand it more then I thought I did!!