Two Teams, Two Slanted Grids

Two survey teams map the same space, but each uses its own slanted axes, its own basis.

• Team North writes a point as coordinates $(c_1, \dots, c_n)$ in its basis $\mathbf{b}_1, \dots, \mathbf{b}_n$, meaning the point is $c_1\mathbf{b}_1 + \dots + c_n\mathbf{b}_n$.
• Team South does the same with its own basis $\mathbf{d}_1, \dots, \mathbf{d}_n$.

Between the two surveys a machine moves every point: in ordinary standard coordinates it sends $\mathbf{x}$ to $M\mathbf{x}$.

North logged each point before the move, in North coordinates. South needs each point after the move, in South coordinates. Every step here is linear, so a single $n \times n$ matrix $T$ does the whole conversion: if $\mathbf{c}$ is a point's North coordinate vector, then $T\mathbf{c}$ is the moved point's South coordinate vector.

Task: write north_to_south(B, D, M) returning $T$ as a list of rows, every entry rounded to 4 decimal places.

• B[k] is North's basis vector $\mathbf{b}_{k+1}$ and D[k] is South's $\mathbf{d}_{k+1}$, each given as a list of $n$ numbers.
• M is an $n \times n$ matrix given as a list of rows.
• Both are genuine bases of $\mathbb{R}^n$ (independent vectors), so every point has exactly one set of coordinates in each.