Exercise 8

A linear system where the right hand side of every equation is 0 is called homogeneous. The augmented matrix of a homogeneous system, for instance, has the following form: $$\left[ \begin{array}{cccc|c} * & * & * & * & 0 \\ * & * & * & * & 0 \\ * & * & * & * & 0 \\ \end{array} \right]$$.

1. Using the concepts we've seen in this section, explain why a homogeneous linear system must be consistent.

2. What values for the variables are guaranteed to give a solution? Use this to offer another explanation for why a homogeneous linear system is consistent.

3. Suppose that a homogeneous linear system has a unique solution.

   1. Give an example of such a system by writing its augmented matrix in reduced row echelon form.

   2. Write just the coefficient matrix for the example you gave in the previous part. What can you say about the pivot positions in the coefficient matrix? Explain why your observation must hold for any homogeneous system having a unique solution.

   3. If a homogeneous system of equations has a unique solution, what can you say about the number of equations compared to the number of variables?