Linear Systems and Networks

Linear Systems and Networks

What are the main facts?

• You had a system of  $n$  equations with  $n$  unknowns that you used to solve using addition method or substitution method.

• Rewrite the system as

   $A\overrightarrow{x}=\overrightarrow{b}$  

• If  $A$  is invertible ( $\det\left(A\right)\ne0$  ), the solution is unique and given by  $\overrightarrow{x}=A^{-1}\overrightarrow{b}$  
  

• If  $A$  is not invertible ( $\det\left(A\right)=0$  ) then you have either an infinite number of solutions or you have no solution.
  

What are the main methods?

• Your main method learned was row reduction.

• Row reduction doesn't require a square matrix.

• Row reduction gives you the result for all cases (unique solution, no solution, and infinite number of solutions.

Step 1 Identify the coefficient matrix and the right hand side, the array (vector) of unknowns and the right hand side.

Step 2: Place this into an Augmented Matrix

 $\left[A\ \ \overrightarrow{b}\right]$  

Step 3 Row Reduce the Matrix

This was initially done by hand-I would review everything.

Extract the "solution"

Solution may be infinite or not exist.

What does it look like if there are an infinite number of solutions?

An example with a system of 2 equations and 2 unknowns

What does it look like if there is no solution?

An example with a system of 2 equations and 2 unknowns.

We used this to study network models.

Inputs=outputs, which yields equations.

What should you remember about those models?

• Bookkeeping: Line up the variables when you set up the system of equations.

• The numbers go on the right.