1. Matrices and Systems Engineering Frank Lipsky copyrighted

2. Definition –Linear Equation Linear Equations A linear equation is defined as an algebraic equation in which each term is either a constant or the product of a variable and a constant, with the variable being restricted to the first power. In other words, you can have a variable such as x (= x 1 ), but not x 2 or x 3, or any other higher power. In addition, linear equations may not include terms that are the product of two variables (such as xy, for example). Linear equations can have one or more variables, although the ones most frequently encountered are probably those with two variables. These variables are often designated as x and y. You will often see a linear equation in two variables written in the form: y = mx + b This notation is called the slope-intercept form. The graph of a linear equation is a straight line (hence the name linear). The constant m will determine the slope of the line (i.e. how steeply it rises from left to right, or vice versa). The constant b determines the y-intercept (the point at which the line crosses the y axis). Linear equations are sometimes referred to as equations of the straight line. Perhaps more important to note is the fact that the slope-intercept form of the equation is sometimes written using different labels for the constants, depending on what part of the world you are in. In Australia, for example, it is written as y = mx + c. Don't let this throw you, as it is the exact same equation. Note that the slope intercept form cannot be used to describe vertical lines, since the slope would be undefined for such a line. It can however describe horizontal lines (m becomes zero, and y is therefore effectively a constant with the same value as b for any value of x). Here is the graph for the linear equation y = 0.5x + 1:

3. Graph of (one) Linear Equation in 2D

4. Graph of (one) linear equation 3D

5. Graph (3) Linear equations

6. Three (3) Linear Equations Matrix Form

7. Closed Loop Control