11.7. Under-determined Systems Homework

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

2. Let,

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

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

   Express the set of solutions to \(\mathbf{A}\,\bm{x} = \bm{b}\) 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)
   

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

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

   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)