Given a vector $(1, 2, 3 )^T$ wrt to the basis B, what are its coordinates wrt to the basis $\alpha$?

Question

I have a vector, $v$, $ (1,2, 3)^T$ wrt to the basis B and I want to find its coordinates wrt to the basis $\alpha$.

The change of basis matrix, $P_2$ from $\alpha$ to B is given by

$$ \begin{pmatrix} 2 & 1 & -1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ \end{pmatrix} $$

The change of basis matrix, $P_2$ from B to $\alpha$ is given by its inverse which is $$ \begin{pmatrix} {1}/{3} & {-2}/{3} & {1}/{3} \\ 0 & 1 & 0 \\ {-1}/{3} & {-1}/{3} & {2}/{3} \\ \end{pmatrix} $$

The solution given on the worksheet uses $P_1$: $P_1 v$.

However I am confused as this is the change of basis matrix from $\alpha$ to B. I thought that we would want to use $P_2$ as this changes basis from B to $\alpha$.

The full pdf can be found below under question 1:

https://math.hawaii.edu/~gautier/change.pdf

Answer 1

We have that

$$v_B=P v_\alpha \iff P^{-1}v_B=P^{-1}P v_\alpha \iff v_\alpha=P^{-1}v_B$$

with

$$P=\begin{pmatrix} 2 & 1 & -1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ \end{pmatrix} \quad P^{-1}=\begin{pmatrix} {1}/{3} & {-2}/{3} & {1}/{3} \\ 0 & 1 & 0 \\ {-1}/{3} & {-1}/{3} & {2}/{3} \\ \end{pmatrix}$$

which agrees with the indication given in the linked paper.

Answer 2

Note what they say in the reminder on the top of page 1: The change of basis matrix from $\alpha$ to $B$ is the matrix $P_1$ which for any vector $v$, with $\alpha$-representation $[v]_\alpha$, fulfills $$ [v]_\alpha=P_1[v]_B $$ So yes, according to that paper, the to and from goes the opposite direction of what you seem to think.