Introduction

Linear Algebra revolves around the concept of solving a system of linear algebraic equations. A linear algebraic equation has the form

a1x1 + a2x2 + ...anxn = b

where: a1...an are real known constants called coefficients, b is a real known constant and x1...xn unknown variables. This equation is algebraic since no derivitives or integrals are present. For the equation to be linear, the x1...xn variables must be first order. The equation is linear since all terms related to the unknowns are of the first order. An example of nonlinear equation is:


3x12 + 2x2 = 5 is nonlinear because the first variable x1 is squared.

A system of equations in which each equation is linear and algebraic and all equations utilize the same variable set is known as a system of linear algebraic equations which can be expressed in matrix form. For simplicity, a system of two linear equations with two unknowns is discussed but the concepts presented can easily be extended to more complicated systems with many more unknowns.

Consider a system of linear algebraic equations that contains two equations as follows:

4x + 2y = 2
2x - y = 2

where this system contains two unknowns x and y. This system of equations can be rewritten in matrix form as


or in compact form as


where  is the coefficient matrix of order (2 x 2) containing known values,  is a column vector of order (2 x 1) containing unknowns and  is a column vector of order (2 x 1) containing known values.

This system of equations can be shown graphically as



These two equations intersected at the point x = 3/4 and y = -1/2. Notice that the values of the intersection point are the only values which satisfy each equation, therefore the intersection point signifies a unique solution.