Algebra variable

Algebra is a subject which helps us to find unknown quantities with minimal information. But how do we carry all mathematical operations when something is not known? We assign letters from alphabet (mostly small case letters) for the unknown.

Such letters are only called as variables or changeable. Why the name is selected as such? Because the value assigned to these could be varied according to our wish but gives the correct information at a required condition. For example I earn $100 per day.

How do I calculate the total earnings after a certain number of days? Here the word certain is really uncertain! In other words it is an unknown. So we assign ‘x’ as the number of days and I can formulate that my total earnings in dollars will be 100x. Now my job is simple. I just to need to replace ‘x’ by the actual number of days I decide and I get the required information correctly.

Thus basically the changeable are employed as algebra variables or simply math variables. But why we had done away with the prefixes ‘algebra’ or ‘math’ and we simply refer as such?
Because, such uses of letters even in other subjects are subjected to mathematical or algebraic operations.
The types of variables are many. In most cases more than one forms are used. For example one may represent an input and another may be the corresponding output. Obviously the former can be assigned any value and hence it is described either as independent forms and the latter, because of the dependence on the input is referred as dependent forms.

In most of the cases they are represented by the letters ‘x’ and ‘y’ respectively. In such cases they are related by equations or functions.
Further, different types of variables are used depending on the context. For example when you study about distance versus time for a moving object, lettert is used for the unknown time and ‘s’ is used for the corresponding distance.

The same letter tis also used for denoting temperatures in normal scale. (whereas, T is used for temperature in absolute scale!). Letters p and v are used for denoting pressure and volume respectively.
We mostly see that the letters at the second half of English alphabet are mostly used for the unknown quantities. It is just a convention and the first of half is generally reserves for constants. However use of letters like ‘a’, ‘b’ ‘c’ is common in geometry.
In addition to English alphabetical letters Greek letters are also extensively used especially in the topics of trigonometry.

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.