# Section 4.7 $\quad$ Homogeneous Systems

Consider the homogeneous system
\begin{equation*}
  A\mathbf{x} = \mathbf{0},
\end{equation*}
where $A$ is $m\times n$, The set of all solutions <br /><br /><br /><br />

### Example 1

Find a basis of the solution space of the linear system
\begin{equation*}
   \left[
    \begin{array}{ccccc}
      1 & 1 & 4 & 1 & 2  \\
      0 & 1 & 2 & 1 & 1  \\
      0 & 0 & 0 & 1 & 2  \\
      1 & -1 &0 & 0 & 2  \\
      2 & 1 & 6 & 0 & 1  \\
    \end{array}
  \right]
  \left[
    \begin{array}{c}
      x_1 \\x_2 \\x_3 \\x_4 \\x_5 \\
    \end{array}
  \right] =
  \left[
    \begin{array}{c}
      0 \\0\\0 \\0 \\0 \\
    \end{array}
  \right]
\end{equation*}


```python
from sympy import *

x1, x2, x3, x4, x5 = symbols('x1 x2 x3 x4 x5');
x = Matrix([x1, x2, x3, x4, x5]);
A = Matrix([[1, 1, 4, 1, 2], [0, 1, 2, 1, 1], [0, 0, 0, 1, 2], [1, -1, 0, 0, 2], [2, 1, 6, 0, 1]]);
Ax = A.dot(x);

solve(Ax)
```




    {x2: -2*x3 + x5, x4: -2*x5, x1: -2*x3 - x5}



**Remark** For the homogeneous system $A\mathbf{x} = \mathbf{0}$, if the reduced row echelon form of $[A~|~\mathbf{0}]$ has $r$ nonzero rows, then
- 
- 

## Relationship between Homogeneous and Nonhomogeneous Systems

For nonhomogeneous system $A\mathbf{x} = \mathbf{b}$, the set of solutions<br /><br /><br /><br />

### Example 2

Consider the linear system
\begin{equation*}
   \left[
    \begin{array}{ccc}
      1 & 2 & -3 \\
      2 & 4 & -6 \\
      3 & 6 & -9 \\
    \end{array}
  \right]
  \left[
    \begin{array}{c}
      x_1 \\x_2 \\x_3
    \end{array}
  \right] =
  \left[
    \begin{array}{c}
      2 \\4\\6  \\
    \end{array}
  \right]
\end{equation*}<br /><br /><br /><br /><br /><br /><br />

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