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.

Dot product

A dot product is an operation that which involves multiplication of two vectors to arrive to a scalar product.  Given two vectors, v=ai + bj and u=ci + dj, v.u read as ‘v dot u’ would be equal to a scalar product, ac + bd. So, basically the product would be a number and not a vector.
The dot product of two vectors would be a scalar even in a three dimensional space, R3.  So, in a three dimensional space given vectors v=ai +bj+ck and u=xi+yj+zk, the dot-product is given by v.w=ax + by +cz. The definition of dot product can be given as the dot product equation of vectors a’ and b’ such that a.b= ax. bx + ay.by = |a||b|cos(theta) .
Here |a| and |b| are the magnitudes of the vectors and theta is the angle between the vectors. It is read as modulus of vector a multiplied with the modulus of vector b, multiplied by the cosine of the angle between the two vectors a’ and b’.
Following are some of the important points to be remembered while finding the scalar product, i.i=1, j.j=1, k.k=1, i.j=0, j.k=0 and k.i=0, this shows that the scalar product of vectors which are perpendicular to each other is zero.
Some of the properties of dot-product are as given below,
• Commutative property: u.v = v.u
• Distributive property: u.(v+w) = (u.v) + (u.w)
• Associative property: (cv). u = v.(cu)= c(u. v)
• 0. u = u.0 = 0
• v.v =|v|2
• If v. v = 0 then v = 0
Let us now take a look at the dot product proof of distributive property given by u. (v+w)=(u.v)+(u.w)
Let the vectors to be, u=(u_1,u_2,u_3...,u_n ); v =(v_1,v_2,v_3...,v_n) and w=(w_1,w_2,w_3...,w_n). On the left hand side we have, u.(v+w) = (u_1,u_2,u_3...,u_n ).[(v_1,v_2,v_3...,v_n)+ (w_1,w_2,w_3...,w_n)]
          =  (u_1,u_2,u_3...,u_n ).[(v_1+w_1), (v_2+w_2), (v_3+w_3)…, (v_n+w_n)] on regrouping we get,
          = [u_1((v_1+w_1), u_2(v_2+w_2), u_3(v_3+w_3),…,u_n  (v_n+w_n)]
Applying the distributive property we get,
= [u_1v_1+u_1w_1, u_2v_2+ u_2w_2, u_3v_3+ u_3w_3….., u_n v_n+ u_n w_n]
Which can be written as, [u_1v_1, u_2v_2, u_3v_3…, u_n v_n] + [u_1w_1, u_2w_2, u_3w_3…, u_n w_n]
On re-writing the above expression we get, [(u_1,u_2,u_3...,u_n ). (v_1,v_2,v_3...,v_n)]+[ (u_1,u_2,u_3...,u_n ). (w_1,w_2,w_3...,w_n)]  which would be the expression on the left hand side, [u.v+u.w] and hence proved!Thus we can prove all the properties using the above computational method.

 u_n it vectors are the vectors with length of one u_n it.  For u_n it vectors u and v, the dot product of u_n it vectors is given by, u.v=cos(theta) where (theta) is the angle between the two u_n it vectors.

Work and Time Calculation

Work and time are two of inter-related concepts in mathematics and science. Work and time related calculations are most often asked in almost all competitive exams. Taught in middle school classes, work and time calculation problems are worked out in SAT, MAT exams as well. The trick is to solve the problems within seconds. Let’s have a look at some of the facts related to work and time calculation in this post.

1. If a person can complete a work in n days, then the person can complete 1/n part of the work in one day. For example: She completed the process of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop in 6 days. Therefore, she will complete 1/6 part of the work of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop.

2. If the number of person to complete a particular work is increased, the time to complete the same work decreases. For example: 100 employees build about 1000 toy action figures in 10 days. If the number of employees is increased to 150, then they will build 1000 toy action figures in less than 10 days because the work is distributed among more workers.

3. If worker A has the capability of working twice as worker B, then A will take ½ of the time that B took to complete a work. For example: B designed the outlook of cot mobile for baby girls in 2 hours. A works twice as B and therefore, A designed the outlook of cot mobile for baby girls in ½ x 2 hours = 1 hour.

These are some of the most important facts to be known while working out work and time calculation in mathematics.  However, the list if not the ultimate one, there are many other such work and time related facts.

Absolute Error

When we do any calculations there are always chances of making mistakes, either we do addition, subtraction or anything, similarly when we measure height, distance or anything with the help of any measuring device there are chances of making a mistake so if we measure the same thing twice we may get different answers and this is due to the error in measuring. Error is not the mistake we have made because it does not give you the wrong answer. The uncertainty in measurement is termed as the error. There are many types of errors which occur in experimental studies.

1. Greatest possible error – This is the error we make when we do the approximation or rounding off to tenth, hundredth place.

2. Absolute Error– This is the error which occurs due to the inaccuracy in the measurement we do. Experimental scientists come across usually with this type of error. This is the amount of physical error we make in the process of measurement. Absolute Error Formula– It is usually denoted by delta x and is equal to difference between the calculated value and the actual value. Now How to Calculate Absolute Error or How to Find Absolute Error– We can find the absolute-error by finding the difference between the inferred value and the calculated value of the measurement. It usually signifies the uncertainty in the measurement process. For example: - If we find the length of stick as 1.09 centimeter though its actual length is 1 centimeter. Then the absolute-error that is delta x = Calculated value – Actual value which is 1.09 – 1 and that is equal to 0.09. Hence we can say that absolute-error is equal to 0.09. Absolute-error is always positive. Therefore we can call it as the absolute value of the difference of the two values which are the calculated value and the actual value.

3. Relative error – This type of error tells you about how good a measurement is relative to size of the thing which is measured. It expresses the ratio of absolute-error to the measurement that is accepted. This actually shows the relative size of the error of the measurement in relation to the measurement itself. The formula for calculating relative error is Relative error = Absolute error over accepted measurement.