Suppose a vector space $V$ has bases $B_1 = \{(1, 0), (0, 1)\}$ and $B_2 = \{(1, 1), (1, -1)\}$ for $\mathbb{R}^2$. What is the change-of-basis matrix $P$ from $B_1$ to $B_2$ (i.e., $[\mathbf{v}]_{B_2} = P[\mathbf{v}]_{B_1}$)?