.. _under_detHW:

Under-determined Systems Homework
===================================

#. Why is there no unique solution to :math:`\mathbf{A}\,\bm{x} = \bm{b}` 
   when the system of linear equations has more unknown variables than
   independent equations.

#. Let,

   .. math::

      \bm{A} = \begin{bmatrix}
                  -8  & 2 & 0 \\ 
                  6  & 0 & -4
               \end{bmatrix} \hspace{0.5in}
      \bm{b} = \begin{bmatrix}
                 -14 \\  20 
               \end{bmatrix}.

   Use the Python ``rref()`` command on an augmented matrix 
   :math:`\left[\mathbf{A} \bm{b}\right]` to show 
   the under-determined augment matrix in row reduced echelon form.

   Express the set of solutions to :math:`\mathbf{A}\,\bm{x} = \bm{b}` as a line
   equation in parametric form.

   ::

        A = np.array([[-8, 2, 0], [6, 0, 4]]).astype(float); print(A)

        b = np.array([[-14, 20]]).T; print(b)

#. Use the Moore-Penrose pseudo-inverse, QR factorization, and
   the ``lstsq`` function to find solutions to the following
   under-determined matrix equation.

   .. math::

      \begin{bmatrix}
         4  & 16 & 8 & 4 \\ 
         8  & 14 & -2 & -4 \\ 
         4  & 5 & -2 & -10
      \end{bmatrix}\begin{bmatrix}
                     x_1 \\ x_2  \\ x_3 \\ x_4 
                   \end{bmatrix} 
              = \begin{bmatrix}
                  1 \\ -7 \\ 4 
                \end{bmatrix}

   ::

        A = np.array([[4, 16, 8, 4],
        [8, 14, -2, -4],
        [4, 5, -2, -10]]).astype(float); print(A)

        b = np.array([[1, -7, 4]]).T; print(b)