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Linear Algebraic Equations
Systems of linear algebraic equations arise in all walks of life. They represent the most basic type of system of equations and they’re taught to everyone as far back as 8-th grade. Yet, the complete story about linear algebraic equations is usually not taught at all. What happens when there are more equations than unknowns or fewer equations than unknowns? These are precisely the questions that are answered below.
Before we processed with this, there’s some background material that we’ll need to learn. We’ll first need to discuss ways to minimize a function of several variables. Then, we’ll need to understand how to do this using a matrix-vector notation. After this is done, we’ll be able to look at linear algebraic equations.
How to Minimize a Function of Several Variables
The best way to introduce this topic is with an example. Let’s minimize the function
(1) [pic]
where x and y are both constrained to lie on the line
(2) [pic]
This problem amounts to finding the point on the line that is closer to the origin than any other point on the line. There are several ways to solve this problem. Let’s look at the following three ways.
The first way to solve this problem is to use geometry. Draw a perpendicular to the line that intersects the origin. The equation of the perpendicular line is y = -x/a. Substituting the equation for the perpendicular line into the equation for the line yields the intercepts
(3) [pic]
The second way to solve this problem is to recognize that this problem is a constrained optimization problem; a problem in which a function is minimized while being subjected to a constraint. The constrained minimization problem is converting into an unconstrained minimization problem. This is done using a substitution step. The constraint, Eq. (2), is substituted into the

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