25.1. System of Linear Equations

Try these first

1. 1.

   What is the general form of a system of linear equations?

   Hint

   Think about the arrangement of equations and their variables.

2. 2.

   What does a homogeneous system mean?

   Hint

   Recall that there's no constant on the right-hand side.

3. 3.

   What is the matrix form of a system of linear equations?

   • Ax = b
   • Ax = 0
   • A + b = x

   Hint

   Recall how we express our equations in one compact form.

4. 4.

   True or False: A system with Rank(A) < Rank([A∨b]) has infinitely many solutions.

   • True
   • False

   Hint

   Think about what inconsistency means in terms of rank.

5. 5.

   Analyze the following system: 3x + 4y - z = 10 2x - 2y + 5z = 5 x - 0.5y + z = 1 Determine if the system is consistent, independent, or dependent, and explain your reasoning.

   Hint

   Consider reducing the system to find its rank and see if the equations give distinct information.

6. 6.

   Given the complex system below, establish its characteristics: 1x + 2y + 3z = 6 4x + 5y + 6z = 15 2x + 4y + 6z = 10 Is it independent, dependent, or inconsistent? Provide rationale.

   Hint

   Look for proportional relationships among the equations to determine if they are independent or not.