def f_b_v3(M):
    return M / M.sum(0)

1. M.sum(0):
   • This sums the array M along axis 0 (the “downward” direction in a 2D matrix).
   • If M has shape $(n, m)$, then M.sum(0) returns a 1D array of length $m$, where each element is the sum of a particular column of M.
2. Division (Broadcasting):
   • When you write M / M.sum(0), numpy’s broadcasting rules automatically “stretch” the 1D array of shape $(m,)$ so that each element in column $j$ of M is divided by the $j$-th value in M.sum(0).
   • In other words, each column of M is divided by its own column sum.
3. Result:
   • The returned 2D array has the same shape as M, but now each column sums to 1.

How is this different from Version 2?

• Version 2 used reshape((1, M.shape[1])) to make the broadcasting dimension explicit.
• In Version 3, you rely on numpy’s automatic broadcasting when dividing by a 1D array of column sums.
• Functionally, both approaches yield the same column-normalised matrix.

Overall, Version 3 is a minimalistic one-liner that leverages numpy’s broadcasting to achieve the same result: each element of M is divided by the sum of its column.