def f_a_v2(m, n):
    M = numpy.ones((m, n))

    x = numpy.arange(m).reshape((m, 1))
    y = numpy.arange(n).reshape((1, n))

    M = M * x
    M = M * y

    return M

Creating an all-ones array

M = numpy.ones((m, n))

• This line initialises a 2D numpy array M with shape (m, n), filled entirely with the value 1.
• After this, M[i, j] is 1 for every i, j.

Generating ranges for row and column indices

x = numpy.arange(m).reshape((m, 1))
y = numpy.arange(n).reshape((1, n))

1. numpy.arange(m)

   Creates a numpy array of integers from 0 to m - 1.

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

2. .reshape((m, 1))

   Reshapes the 1D array into a 2D array with m rows and 1 column.

   For m = 3, x becomes:

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

   So, x is shape (3, 1).

3. numpy.arange(n).reshape((1, n))

   • Similarly, this creates an array of integers from 0 to n - 1, then reshapes it into one row and n columns.
   • For n = 4, y becomes:

   $$ [0\ 1\ 2\ 3] $$

   So, y has shape (1, 4).

Elementwise multiplication with broadcasting

M = M * x
M = M * y

• First multiplication: M = M * x

  Since M was all ones of shape (m, n), and x is shape (m, 1), the multiplication is done via numpy broadcasting.

  • Broadcasting means that x is “stretched” or “replicated” across all the columns of M.
  • Effectively, each row i in M gets multiplied by x[i, 0].
  • After this step, M[i, j] = i for all valid i, j.

    • For example, if m = 3 and n = 4, after the first multiplication:

      M = [[0, 0, 0, 0], # row i = 0 (0 * 1 = 0 for all columns)
           [1, 1, 1, 1], # row i = 1 (1 * 1 = 1 for all columns)
           [2, 2, 2, 2]] # row i = 2 (2 * 1 = 2 for all columns) 
      

• Second multiplication: M = M * y

  Now M is multiplied by y which has shape (1, n). This time, the array y is broadcasted along each row.

  • For each column j, the multiplication factor becomes y[0, j] = j.

  • After this step, each element in M becomes i * j.

  • Continuing the example with m = 3 and n = 4:

    M = [[0, 0, 0, 0], # row i = 0 multiplied by each column index j
         [0, 1, 2, 3], # row i = 1 multiplied by each column index j
         [0, 2, 4, 6]] # row i = 2 multiplied by each column index j
    

Hence, the result is a 2D array of shape (m, n) whose element at (i, j) is i * j.

Final return