Linear Transformation

Let us try to give a simple introduction and explanation about linear transformations. Let us not scare the readers with hi-fi terms like ‘vector spaces’ ‘matrices’ and symbols like ‘e’ ‘Rn’ etc. Yes, let us involve those at a higher level after getting acquainted with what basically a linear-transformation means.

In a data different scores, that is, items described by numbers are exhibited. There is a possibility to express the items of the data in the form of a pattern. If such a pattern is in the linear form that the transformation of the data set to a pattern is called as linear transformation. It may be noted that such a transformation can be fairly accurate for a limited interval meaning limited number of items in the data set.

This type of transformation is obviously results as a linear function in the form a + bx (similar to mx + b in analytical geometry). Since linear functions are always ‘one to one’, this type of transformation is also referred as one to one linear transformation. Recording the scores of student in a class can be cited as one of the linear transformation examples.

Let us discuss about the basic concept of this type of transformation. Suppose Xi represents the item in general, of the given data, and if X’I is the same after the transformation of the data, then the linear relation is X’ = a + b Xi, where a and b are constants for the particular transformation. The letter a is called additive component and b is the multiplicative component of transformation.

These are analogous to y-intercept and slope of linear algebraic functions. One must know what should be mean and standard deviation of the transformed data and accordingly the values of constants a and b are determined. Because the condition of a linear-transformation is X’m =   a + b Xm and X’s = b Xs, where the subscripts m and s refer the respective mean and standard deviation.

Let us illustrate a linear transformation example. Suppose a data is describes the scores as 13, 16, 21, 21, 24. This has to be linearly transformed with a mean of 95 and standard deviation of 15. What is the formula of transformation?

The mean and the standard deviation of the given data are Xm = 19 and Xs = 4.42, rounded to nearest hundredth. The set of desired figures in the transformation is X’m = 95 and X’s = 15. Since, X’s = b Xs,
b = 3.39 and a = 95 - 3.39*19 = 30.59, rounded to two decimal place.
Thus the transform relation is X’ = 30.59 + 3.39X.

Linear Programming with the Help Of Simplex Algorithm

The concept of programming is very important. It is being used in the field of mathematics as well. Linear programming is a very important concept and is now very widely used in the field of mathematics. The Simplex method tutorial is a part of the linear programming model. This method is also called an algorithm. This algorithm is used as part of linear programming. This is used in finding a optimal solution.

The Simplex method examples can be very helpful in understanding and knowing more about this algorithm. For understanding this method a geometric figure called the polytope has to be studied. Basically a polygon is a geometric figure which has many sides. So, hexagon is a geometric figure which has six sides.

A pentagon is a geometric figure which has five sides. Similarly there are other geometric figures which have different number of figures and they are given various names. In Simplex methods the polytope plays a very important role as this gives the area which is under consideration for finding the optimal solution. So, this concept has to be learnt properly.

There is different number of vertices present in a polytope. To find the optimal solution, the process begins from any one of the vertices of the polytope and moves towards the vertex which shows the optimal solution. This can be represented in a standard form.

Another form can be used in this case, namely the canonical form. There are two methods that can be used. The two methods are called the M-method and the other one is called two-phase method. As the name suggests in the two-phase method there are two phases that are to be considered to arrive at the final solution.

The final solution is nothing but the optimal solution. The ultimate purpose is to arrive at the optimal solution. An example can be used to explain the concept. An equation will be given for simplification. There will also be some constraints given. The simplification has to be done keeping these constraints in mind.

The constraints can also be in the form of equations. These equations must be taken into account while performing the simplification procedure. Then they can be represented in the canonical form and a feasible solution is found for the variables present in the equation, keeping in mind the constraints given. Once this is done the optimal solution is found out.