.. _over_detHW:

Over-Determined Systems Homework
==================================

#. When does  the vector projection solution to an over-determined 
   system of matrix equations give an exact solution to
   :math:`\mathbf{A}\,\bm{x} = \bm{b}`?

#. Notice that this problem has more equations than unknowns. Still,
   form the :math:`\bf{A}` and :math:`\bm{b}` matrices in Python and
   compute the ranks of :math:`\bf{A}` and 
   :math:`\left[\mathbf{A}\,\bm{b}\right]`. Is
   :math:`\bm{b}` in the column space of :math:`\bm{A}`? Can we find an
   exact solution, or only an approximate solution?

   .. math::

      \begin{bmatrix}
         15  & -3 & 15 \\ 
         10  & -6 & -7 \\ 
         5  & 8 & 1 \\ 
         0  & 6 & 2
      \end{bmatrix}
      \begin{bmatrix}
        x_1 \\  x_2 \\  x_3 
      \end{bmatrix} =
      \begin{bmatrix}
        24 \\  -9 \\  22 \\  14 
      \end{bmatrix}

   ::

        A = np.array([[15, -3, 15],
        [10, -6, -7],
        [5, 8, 1],
        [0, 6, 2]]).astype(float); print(A)

        b = np.array([[24, -9, 22, 14]]).astype(float).T; print(b)

#. For the previous system of equations, use Python to find
   the solution from the projection equation and using the
   QR factorization.

#. For the previous :math:`\bf{A}` matrix, use Python to find its
   pseudo-inverse, :math:`\mathbf{A}^+`. Verify that 
   :math:`\bm{x} = \mathbf{A}^+\,\bm{b}` finds the same result as found
   from the projection equation and from QR.