Well I have the following problem: Let $\alpha = \{v_1,v_2,v_3\}$ and $\beta=\{u_1,u_2,u_3\}$ be two bases of $\mathbb{R}^3$ such that $v_1=(1,0,1)$, $v_2=(1,1,0)$ and $v_3=(0,1,1)$. It's known that if we denote $[I]^\alpha_\beta$ the matrix of the identity map in the basis $\alpha$ and $\beta$ we have:
$$[I]^\alpha_\beta=\begin{pmatrix}1 & 2 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 0\end{pmatrix}$$
Find $u_1,u_2,u_3$ in the cannonical basis. My approach was the following: let $\epsilon = \{e_1,e_2,e_3\}$ be the cannonical basis of $\mathbb{R}^3$ and let $[I]^\beta_\epsilon$ be the change of basis matrix from $\beta$ to $\epsilon$ and $[I]^\alpha_\epsilon$ be the change of basis matrix from $\alpha$ to $\epsilon$, then we have the relation:
$$[I]^\alpha_\epsilon = [I]^\beta_\epsilon [I]^\alpha_\beta$$
And we want to find $[I]^\beta_\epsilon$, so we compute the inverse $([I]^\alpha_\beta)^{-1}$ and then $[I]^\beta_\epsilon = [I]^\alpha_\epsilon ([I]^\alpha_\beta)^{-1}$ and we can get the $u_i$ as the columns of this matrix.
Now, is there another better way to find these vectors?
Thanks very much in advance!