.. _syseqHW:

Systems of Equations Homework
==============================

#. For the following system of equations, express the equations as a
   matrix equation of the form :math:`\mathbf{A\,}\bm{x} = \bm{b}`.

   .. math::

      \left\{\begin{aligned}
        7x_1 + 9x_2 + 4x_3 = 102 \\ 
        17x_1 + 8x_2 + 6x_3 = 171 \\ 
        -7x_1 + -7x_2 + 10x_3 = -64 
      \end{aligned}\right.

#. Use the `solve` function from Python's `scipy.linalg` module to find the 
   solution to the previous system of equations.


#. Apply elimination to the 2-by-3 augmented matrix
   :math:`[\mathbf{A}\:\bm{b}]`. What is the triangular system
   :math:`\mathbf{U}\,\bm{x} = \bm{c}`? What is the solution
   :math:`\bm{x}`?

   .. math::

      \mathbf{A}\,\bm{x} = \begin{bmatrix}
                              2  & 3 \\ 
                              4  & 1
                           \end{bmatrix}
      \begin{bmatrix}
        x_1 \\  x_2 
      \end{bmatrix} =
      \begin{bmatrix}
        1 \\  17 
      \end{bmatrix}

#. For each system of equations, use Python’s ``numpy.linalg.matrix_rank``
   function to determine if a solution exists. Note the rank of
   :math:`\bf{A}`. If a solution exists, use Python’s `scipy.linalg.solve`
   function to find it.

   a. 

      .. math::

         \left\{\begin{aligned}
           x_1 + 2x_2 + 2x_3 = 1 \\ 
           4x_1 + 8x_2 + 9x_3 = 3 \\ 
           3x_2 + 2x_3 = 1 
         \end{aligned}\right.

   b. 

      .. math::

         \left\{\begin{aligned}
           x_1 + 2x_2 + 3x_3 = 1 \\
           3x_2 + x_3 = 2 \\ 
           x_1 + 14x_2 + 7x_3 = 3 
         \end{aligned}\right.

   c. 

      .. math::

         \left\{\begin{aligned}
           7x_1 + 7x_2 = 35 \\ 14x_1 + 7x_2 = 49 
         \end{aligned}\right.