So here comes an example, suppose I have a problem like this. So you have some negative right hand side, you have some weird variables and you have inequalities. That say how may we convert that into a standard form. So, what we want to do first is to change the right hand side, okay? So we flip the sign for the second constraint, that's how you here see negative 1. And you may see that this negative 2 does not change, why is that? It's simply because your x2 originally is non positive. So later we will convert it to a non negative 1 and that's how your x2 is flipped again. So these positive things all becomes negative, right. And you also need to fix your x3 because it's unrestricted inside. So your x3 becomes x3 minus x4, x3 becomes x3 minus x4 and so on. Lastly, once you do all the things, you would get a greater than or equal to a less than or equal to. So here you minus us like here, you add us like, that's how you make these standard form. So we may in many cases also express a standard form linear program in matrixes, okay. With matrixes we express our standard form in this way, so for example, if you have this particular standard form okay? Equality, non negative right hand side and non negative variables. In that case we may just expressed this problem by defining c, a and b, okay? So your c is your coefficient of objective functions here you actually need four values, right. Because you have four variables, so you may see that our c vector is two negative one, okay to negative 100. So c is your objective coefficient vector, b is your right hand side vector, so it's five and four and then lastly, a is your constraint coefficient matrix. So it's 1, 3, 5 negative 6 is here. And for this part it is an identity matrix, all right. So this is nothing but using a more compact, more abbreviated way to write down a formulation okay. The general form is always in this way, we want to minimize c transpose x subject to Ax equals P and x non negative. All you need to do is to specify the values for your coefficient matrix for the right hand side vector and for your objective vector, okay? An objective function can be either max or min, doesn't really matter. It doesn't really matter whether you have max or min in a standard form. Lastly, their dimensions must of course, be consistent. So we always say we have m constraints, n variables, so that's why you're a matrix is always m by n, okay? So that's pretty much a way to get a standard form. So now, all we need to do is to find a way to solve standard form linear programs, okay? So all we need to do is to say, okay, if you have a problem, let's make it a standard form. So if this is a standard form linear program, and we are able to find its optimal solution, we always is able to go back to solve our original problem by just some simple arithmetic, right? But the difficult part is that, how may we solve a standard form? Well, a standard form linear program is still a linear program right? So it has an optimal solution, it must still have an extreme point optimal solution. So we will find find a way to search among extreme points, that's pretty much our idea. And then later we will show you how may we use algebra to define extreme points.
