def f_a_v3(m, n):
    return numpy.arange(m).reshape((m, 1)) * numpy.float64(numpy.arange(n))

Creating a column vector for row indices

numpy.arange(m).reshape((m, 1))

• numpy.arange(m) produces an array of integers from 0 up to m - 1.

  For example, if m = 3, numpy.arange(m) is [0, 1, 2].

• .reshape((m, 1)) turns this 1D array into a 2D array with shape (m, 1).

  Continuing our example, this becomes:

  $$ \begin{bmatrix} 0\\ 1\\ 2 \end{bmatrix} $$

  This gives you a “column” of indices, one for each row.

Creating a row vector for column indices

numpy.float64(numpy.arange(n))

• numpy.arange(n) produces an array of integers from 0 to n - 1.

  For example, if n = 4, that is [0, 1, 2, 3].

• numpy.float(…) casts the resulting array to type float64. Technically, you could also rely on numpy’s automatic type promotion, but explicitly converting it ensures the final result will be floating point.

• Because this is not further reshaped, it remains a 1D array (shape(n,)). You can think of it as a row of column indices.

Broadcasting and multiplication

... * ...

• When you multiply a (m, 1) array by a (n,) array, numpy broadcasts them together.

  • The (m, 1) array effectively replicates the single column across all columns of length n.
  • The (n,) array replicates its single row across all rows of length m.

• As a result, each element of the final 2D array at position (i, j) becomes:

  $$ i \times j $$

  in floating point format.

Returning the result

The function returns this broadcasted product immediately. If you pass m = 3 and n = 4, the returned 2D array (of shape (3, 4)) is:

[[0. 0. 0. 0.]
 [0. 1. 2. 3.]
 [0. 2. 4. 6.]]