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: