Same predictions, different coefficients

Take three deliberately synthetic observations, x = [1, 2, 3]. Duplicate the feature to make A = [x, x] and set y = 2*x. Coefficients [2, 0] and [0, 2] both reproduce y, because every row depends only on the sum of the coefficients. These observations cannot identify the separate effects.

pinv(A) @ y      # [1, 1]

All exact solutions have the form [1+t, 1-t], with squared norm 2 + 2*t**2, smallest at t = 0. So pinv picks [1, 1]. Minimum norm is a mathematical selection rule; it does not establish equal real effects. With noisy observations, pinv picks the minimum-norm solution among the least-squares minimizers.