Problema de canvi de base en espais vectorials

En l’espai vectorial $\mathbb{R}^3$, es consideren dues bases: $$B_1 = {\mathbf{u}_1 = (1,0,1), \mathbf{u}_2 = (1,1,0), \mathbf{u}_3 = (0,0,1)}$$ $$B_2 = {\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}$$ Si la matriu de canvi de base, prenent com a nova base la base $B_2$ i com a base antiga la base $B_1$, és: $$P = \begin{pmatrix}1 & 1 & 2 \\ 2 & 1 & 1 \\ -1 & -1 & 1 \end{pmatrix}$$ Podem calcular els vectors de la base $B_2$?

Les columnes de la matriu $P$ són les coordenades en la base antiga dels vectors de la base nova, és a dir,

\begin{equation}
\mathbf{v}_1 = \mathbf{u}_1 + 2 \mathbf{u}_2 – \mathbf{u}_3 = (1,0,1) + 2(1,1,0) – (0,0,1) = (3,2,0)
\end{equation}

\begin{equation}
\mathbf{v}_2 = \mathbf{u}_1 + \mathbf{u}_2 – \mathbf{u}_3 = (1,0,1) + (1,1,0) – (0,0,1) = (2,1,0)
\end{equation}

\begin{equation}
\mathbf{v}_3 = 2 \mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 = 2(1,0,1) + (1,1,0) + (0,0,1) = (3,1,3)
\end{equation}