25.5. General Form of Solutions

Try these first

1. 1.

   What is an example of a homogeneous system?

   Hint

   Look for equations equal to zero.

2. 2.

   What does the null space refer to?

   Hint

   Consider the solution space of homogeneous equations.

3. 3.

   What is a homogeneous system?

   • A system with b=0
   • A system with b≠0
   • A system with infinite solutions

   Hint

   Think of the equation forms.

4. 4.

   True or False: A trivial solution exists in all linear systems.

   • True
   • False

   Hint

   Reflect on when the solutions equal zero.

5. 5.

   Given the matrix A, determine the general form of the solution to the equation Ax=0 where Rank(A)=2 and n=4.

   Hint

   Consider the relationship between rank and nullity.

6. 6.

   For a non-homogeneous system given by Ax=b, describe how you would find the complete set of solutions.

   Hint

   Break down into finding individual parts.