# Linear Simultaneous Equations

The general form of a set of linear simulatenous equations:

This can be rewritten in a matrix/vector form:

Equations of this form have three cases for their solutions:

• The system has no solution
• The system has a unique solution
• The system has an infinite number of solutions
• can take a number of values

An over-determined system has more equations than unknowns and has no solution:

An under-determined system has more unknowns than equations and has infinite solutions:

A consistent system has a unique solution

The solution for this system is . Note that the rank and order of are both 2, and exists in this case. If the determinant of a consistent system is 0, there will be no solutions.

## Solutions of Equations

To determine which of the three cases a system is:

• Introduce the augmented matrix:
• Calculate the rank of and

### No Solution

• If , then the system has no solution
• All vectors will result in an error vector
• A particular error vector will minimise the norm of the equation error
• The least square error solution,

### Unique Solution

where is the number of variables in

• .

### Infinite Solutions

• Paramaeters can be assigned to any elements of the vector and the remaining elements can be computed in terms of these parameters
• A particular vector will again minimise the square of the norm of the solution vector

## Homogenous Systems

A system of homogenous equations take the form:

• is an x matrix of known coefficients
• is an x null column vector
• is an x vector of unknowns

The augmented matrix and , so there is at least one solution vector . There are two possible cases for other solutions:

• and , then the trivial solution is the only unique solution
• If and , then there is an infinite number of non-trivial solutions
• This includes the trivial solution

## Example 1

Solutions to:

First calculate the determinant of :

so is a full rank matrix (rank = order = 3). We know solutions exist, but need to find the rank of to check if unique or infinite solutions. Using gaussian elimination to put into row-echelon form:

The rank of , so there is a unique solution

## Example 2

Solutions to:

There is the trivial solution , but we need to known if there is infinite solutions, which we can determine from . Putting it into row-echelon form:

, so there is infinite solutions. Can introduce a parameter to express solutions in terms of. Using the coefficients from the row-echelon form: