25.5.1. Homogeneous Systems

Try these first

1. 1.

   What defines a homogeneous system of equations?

   Hint

   Consider the form of the system.

2. 2.

   What is the trivial solution to a homogeneous system?

   Hint

   Think about the simplest solution.

3. 3.

   What is a homogeneous system of linear equations?

   • A system where Ax = 0
   • A system where b ≠ 0
   • A system with unique solutions

   Hint

   Recall the definition we discussed.

4. 4.

   True or False: The trivial solution of a homogeneous system is x = 1.

   • True
   • False

   Hint

   Think about the simplest solution again.

5. 5.

   Given a 3x3 matrix A with rows that are linearly dependent, find the null space of A and discuss the implications on the solutions of Ax = 0.

   Hint

   Look for combinations of rows that demonstrate linear dependency.

6. 6.

   If a homogeneous system Ax = 0 has three variables and the rank of A is 1, describe the geometric interpretation of the solution space.

   Hint

   Draw the axes and think about which direction represents the span of the solution.