Mathematics :

Mathematics is the abstract study of topics such as quantity (numbers),[structure,space,and change.There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Mathematicians seek out patterns and use them to formulate new conjectures.

Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Mathematics  is a science that developed from the investigation of geometric figures and the computing with numbers. For mathematics, there is no commonly accepted definition; today it is usually described as a science that investigates abstract structures that it created itself by logical definitions using logic for their properties and patterns. This is much worse, as it portrays mathematics as a subject without any contact to, or interest from, a real world.

Even humorous versions of such “distinguishing statements” such as ◦ “Mathematics is the part of physics where the experiments are cheap.” ◦ “Mathematics is the part of philosophy where (some) statements are true — without debate or discussion.” ◦ “Mathematics is computer science without electricity.” (So “Computer science is mathematics with electricity.”) contain a lot of truth and possibly tell us a lot of “characteristics” of our subject. None of these is, of course, completely true or completely false, but they present opportunities for discussion.

What we do in mathematics?

We could also try to define mathematics by “what we do in mathematics”: This is much more diverse and much more interesting than the Wikipedia descriptions! Could/should we describe mathematics not only as a research discipline and as a subject taught and learned at school, but also as a playground for pupils, amateurs, and professionals, as a subject that presents challenges (not only for pupils, but also for professionals as well as for amateurs), as an arena for competitions, as a source of problems, small and large, including some of the hardest problems that science has to offer, at all levels from elementary school to the millennium problems?

Mathematics is based on deductive reasoning though man's first experience with mathematics was of an inductive nature. This means that the foundation of mathematics is the study of some logical and philosophical notions. We elaborate in simple terms that the deductive system involves four things:

(1) A set of primitive undefined terms;

(2) Definitions evolved from the undefined terms; (

3) Axioms or postulates;

(4) Theorems and their proofs. We also include some historical remarks on the nature of mathematics. http://www.publicscienceframework.org/journal/allissues/7040

<br />Mathematics<br />noun <br />1. the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. <br />2. mathematical procedures, operations, or properties.<br />

Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

Linear Algebra:

Linear algebra is a field of mathematics that is universally agreed to be a prerequisite to a deeper understanding of machine learning.Although linear algebra is a large field with many esoteric theories and findings, the nuts and bolts tools and notations taken from the field are practical for machine learning practitioners. With a solid foundation of what linear algebra is, it is possible to focus on just the good or relevant parts.

Linear algebra is a branch of mathematics, but the truth of it is that linear algebra is the mathematics of data. Matrices and vectors are the language of data.Linear algebra is about linear combinations. That is, using arithmetic on columns of numbers called vectors and arrays of numbers called matrices, to create new columns and arrays of numbers. Linear algebra is the study of lines and planes, vector spaces and mappings that are required for linear transforms.

It is a relatively young field of study, having initially been formalized in the 1800s in order to find unknowns in systems of linear equations. A linear equation is just a series of terms and mathematical operations where some terms are unknowny = 4 * x + 1.

Equations like this are linear in that they describe a line on a two-dimensional graph. The line comes from plugging in different values into the unknown x to find out what the equation or model does to the value of y.The column of y values can be taken as a column vector of outputs from the equation. The two columns of floating-point values are the data columns, say a1 and a2, and can be taken as a matrix A. The two unknown values x1 and x2 can be taken as the coefficients of the equation and together form a vector of unknowns b to be solved.

This gives a small taste of the very core of linear algebra that interests us as machine learning practitioners. Much of the rest of the operations are about making this problem and problems like it easier to understand and solve.Problems of this form are generally challenging to solve because there are more unknowns (here we have 2) than there are equations to solve (here we have 3). Further, there is often no single line that can satisfy all of the equations without error. Systems describing problems we are often interested in (such as a linear regression) can have an infinite number of solutions.

Linear algebra is a field of mathematics that is universally agreed to be a prerequisite to a deeper understanding of machine learning.Although linear algebra is a large field with many esoteric theories and findings, the nuts and bolts tools and notations taken from the field are practical for machine learning practitioners. With a solid foundation of what linear algebra is, it is possible to focus on just the good or relevant parts.you will discover what exactly linear algebra is from a machine learning perspective.

After completing this tutorial, you will know:

• Linear algebra is the mathematics of data.
• Linear algebra has had a marked impact on the field of statistics.
• Linear algebra underlies many practical mathematical tools, such as Fourier series and computer graphics.   

Linear algebra is a branch of mathematics, but the truth of it is that linear algebra is the mathematics of data. Matrices and vectors are the language of data.Linear algebra is about linear combinations. That is, using arithmetic on columns of numbers called vectors and arrays of numbers called matrices, to create new columns and arrays of numbers. Linear algebra is the study of lines and planes, vector spaces and mappings that are required for linear transforms.Equations like this are linear in that they describe a line on a two-dimensional graph. The line comes from plugging in different values into the unknown x to find out what the equation or model does to the value of y.

The column of y values can be taken as a column vector of outputs from the equation. The two columns of floating-point values are the data columns, say a1 and a2, and can be taken as a matrix A. The two unknown values x1 and x2 can be taken as the coefficients of the equation and together form a vector of unknowns b to be solved.Problems of this form are generally challenging to solve because there are more unknowns (here we have 2) than there are equations to solve (here we have 3). Further, there is often no single line that can satisfy all of the equations without error. Systems describing problems we are often interested in (such as a linear regression) can have an infinite number of solutions.Some clear fingerprints of linear algebra on statistics and statistical methods include:

• Use of vector and matrix notation, especially with multivariate statistics.
• Solutions to least squares and weighted least squares, such as for linear regression.
• Estimates of mean and variance of data matrices.
• The covariance matrix that plays a key role in multinomial Gaussian distributions.
• Principal component analysis for data reduction that draws many of these elements together.

As you can see, modern statistics and data analysis, at least as far as the interests of a machine learning practitioner are concerned, depend on the understanding and tools of linear algebra.

Applications of Linear Algebra:

As linear algebra is the mathematics of data, the tools of linear algebra are used in many domains.Gilbert Strang provides a chapter dedicated to the applications of linear algebra. In it, he demonstrates specific mathematical tools rooted in linear algebra. Briefly they are:

• Matrices in Engineering, such as a line of springs.
• Graphs and Networks, such as analyzing networks.
• Markov Matrices, Population, and Economics, such as population growth.
• Linear Programming, the simplex optimization method.
• Fourier Series: Linear Algebra for functions, used widely in signal processing.
• Linear Algebra for statistics and probability, such as least squares for regression.
• Computer Graphics, such as the various translation, rescaling and rotation of images.

Another interesting application of linear algebra is that it is the type of mathematics used by Albert Einstein in parts of his theory of relativity. Specifically tensors and tensor calculus. He also introduced a new type of linear algebra notation to physics called Einstein notation, or the Einstein summation convention.

What is a Matrix?

This article introduces the matrix - the rectangular array at the heart of matrix algebra. Matrix algebra is used quite a bit in advanced statistics, largely because it provides two benefits.

• Compact notation for describing sets of data and sets of equations.
• Efficient methods for manipulating sets of data and solving sets of equations.

Matrix Definition:

A matrix is a rectangular array of numbers arranged in rows and columns. The array of numbers below is an example of a matrix.Numbers that appear in the rows and columns of a matrix are called elements of the matrix. In the above matrix, the element in the first column of the first row is 21; the element in the second column of the first row is 62; and so on.The number of rows and columns that a matrix has is called its ]or its order. By convention, rows are listed first; and columns, second. Thus, we would say that the dimension (or order) of the above matrix is 3 x 4, meaning that it has 3 rows and 4 columns.

Matrix Equality:

To understand matrix algebra, we need to understand matrix equality. Two matrices are equal if all three of the following conditions are met:

• Each matrix has the same number of rows.
• Each matrix has the same number of columns.
• Corresponding elements within each matrix are equal.

How to Add and Subtract Matrices?

Two matrices may be added or subtracted only if they have the same dimention; that is, they must have the same number of rows and columns.Addition or subtraction is accomplished by adding or subtracting corresponding elements.

Matrix Multiplication:

Multiply a Matrix by a Number:

In matrix algebra, there are two kinds of matrix multiplication: multiplication of a matrix by a number and multiplication of a matrix by another matrix.When you multiply a matrix by a number, you multiply every element in the matrix by the same number. This operation produces a new matrix, which is called a scalar multiple.

 Multiply a Matrix by a Matrix:

The matrix product AB is defined only when the number of columns in A is equal to the number of rows in B. Similarly, the matrix product BA is defined only when the number of columns in B is equal to the number of rows in A.

Suppose that A is an i x j matrix, and B is a j x k matrix. Then, the matrix product AB results in a matrix C, which has irows and k columns; and each element in C can be computed according to the following formula.

                                          C_ik = Σ_j A_ijB_jk

            where

                    C_ik = the element in row i and column k from matrix C
                    A_ij = the element in row i and column j from matrix A
                   B_jk = the element in row j and column k from matrix B
                   Σ_j = summation sign, which indicates that the a_ijb_jk terms should be summed over j

Multiplication Order:

As we have already mentioned, in some cases, matrix multiplication is defined for AB, but not for BA; and vice versa. However, even when matrix multiplication is possible in both directions, results may be different. That is, AB is not always equal to BA.

Because order is important, matrix algebra jargon has evolved to clearly indicate the order in which matrices are multiplied.

• To describe the matrix product AB, we can say A is postmultiplied by B; or we can say that B is premultiplied by A.
   
• Similarly, to describe the matrix product BA, we can say B is postmultiplied by A; or we can say that A ispremultiplied by B.

Vector Multiplication:

The multiplication of a vector by a vector produces some interesting results, known as the vector inner product and as the vector outer product.

Vector Inner Product:

Assume that a and b are vectors, each with the same number of elements. Then, the inner product of a and b is s.

                            a'b = b'a = s

                         where
                                a and b are column vectors, each having n elements,
                                 a' is the transpose of a, which makes a' a row vector,
                                b' is the transpose of b, which makes b' a row vector, and
                                s is a scalar; that is, s is a real number - not a matrix.

Vector Outer Product:

Assume that a and b are vectors. Then, the outer product of a and b is C.

                          ab'= C

where
      a is a column vector, having m elements,
      b is a column vector, having n elements,
      b' is the transpose of b, which makes b' a row vector, and
      C is a rectangular m x n matrix

Elementary Matrix Operations:

Elementary matrix operations play an important role in many matrix algebra applications, such as finding the inverse of matrix and solving simultaneous linear equations.

Elementary Operations:

There are three kinds of elementary matrix operations.

1. Interchange two rows (or columns).
2. Multiply each element in a row (or column) by a non-zero number.
3. Multiply a row (or column) by a non-zero number and add the result to another row (or column).

When these operations are performed on rows, they are called elementary row operations; and when they are performed on columns, they are called elementary column operations.

Elementary Operators:Each type of elementary operation may be performed by matrix multiplication, using square matrices calledelementary operators.

How to Perform Elementary Row Operations:

     To perform an elementary row operation on a A, an r x c matrix, take the following steps.

1. To find E, the elementary row operator, apply the operation to an r x r identity matrix.
2. To carry out the elementary row operation, premultiply A by E.

How to Perform Elementary Column Operations?

To perform an elementary column operation on A, an r x c matrix, take the following steps.

1. To find E, the elementary column operator, apply the operation to an c x c identity matrix.
2. To carry out the elementary column operation, postmultiply A by E.

Uses of matrix:

Matrices are used in the study of electrical circuits, quantum mechanics and optics. Engineers use matrices to model physical systems and perform accurate calculations needed for complex mechanics to work.Matrices are used much more in daily life than people would have thought. In fact it is in front of us every day when going to work, at the university and even at home.Graphic software such as Adobe Photoshop on your personal computer uses matrices to process linear transformations to render images. A square matrix can represent a linear transformation of a geometric object.

For example, in the Cartesian X-Y plane, the matrix reflects an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of an assassin reflected in a pond of blood. If the video game has curved reflecting surfaces, such as a shiny metal shield, the matrix would be more complicated, to stretch or shrink the reflection.

In physics related applications, matrices are used in the study of electrical circuits, quantum mechanics and optics. Engineers use matrices to model physical systems and perform accurate calculations needed for complex mechanics to work. Electronics networks, airplane and spacecraft, and in chemical engineering all require perfectly calibrated computations which are obtained from matrix transformations. In hospitals, medical imaging, CAT scans and MRI’s, use matrices to operate.Whereas in programming which is taught at the university, matrices and inverse matrices are used for coding and encrypting messages. A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving.

In robotics and automation, matrices are the basic components for the robot movements. The inputs for controlling robots are obtained based on the calculations from matrices and these are very accurate movements.Many IT companies also use matrices as data structures to track user information, perform search queries, and manage databases. In the world of information security, many systems are designed to work with matrices. Matrices are used in the compression of electronic information, for example in the storage of biometric data in the new Identity Card in Mauritius.

In geology, matrices are used for making seismic surveys. They are used for plotting graphs, statistics and also to do scientific studies and research in almost different fields. Matrices are also used in representing the real world data’s like the population of people, infant mortality rate, etc. They are best representation methods for plotting surveys. In economics very large matrices are used for optimization of problems, for example in making the best use of assets, whether labour or capital, in the manufacturing of a product and managing very large supply chains.