Empty set

An empty set, as obviously indicates, is a set which is empty. In other words in an empty-set there are no elements. Sometimes it is also referred as null set. An empty set symbol can be any of the followings.      { } , (/),Ø. Of all these, the last symbol is very prominent.

It may be noted that it is not same as the Greek letter F. There is an interesting point in referring about empty-set. A set when empty is in a way common to all the sets that we think of. Therefore a set with no elements is better referred as ‘the empty set’, the word ‘the’ is more apt than the word ‘an’.

The null set, apart from the fact of having no elements has other properties too. The empty set can be a subset of any set. This may be a bit difficult to understand but a Venn diagram will be helpful to understand clearly. The union of the null set with any set A is the set A itself. It is obvious because we add only a 0 to the number of elements of A.

The intersection of the null set with any set A is again the null set. It is very clear that the intersection set cannot have any elements and it has to be empty. There can be only one subset of the null set which is the null set itself.

Now let us discuss about power set of the empty set. Before that let us see what a power set is. A power set of a set A is the set of all possible subsets of set A, including the null set. For example, if A is {a, b, c}, then all the possible subsets are { }, {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}. Therefore, the power set AP(A) is defined as {{ }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}. Thus we find there are 8 subsets for a set A having three elements.

That is, 23 subsets for 3 elements and in set language the power set P(A) in this case can be expressed as P(A) = 23.  Extending this concept in general for a set A having ‘n’ number of elements, P(Ø) = 2n. Therefore, in case of a null set the power set of the null set is P(A) = 20 = 1. That is the null set itself. In other words, P(Ø) = { }.

Learning About the Field of Vectors

The quantities in mathematics can have different features. The features can be magnitude and sometimes direction as well. All the quantities do not have the same feature. This difference in features can be the difference between the quantities. The quantities that have only magnitude are known to be scalar quantities and the quantities that have both the magnitude and a direction can be called vectors or carriers.

This is the basic difference between the two types of quantities. This difference must be understood to understand the difference between the two types of quantities. This difference can be very helpful in understanding the difference between the quantities. The vector space over a field can be understood only when the concept of carriers is clear and understood. It is part of a mathematical in which the carriers which can be multiplied with a scalar.

The operation of addition is also possible within this space. The scalars can be any numbers. Sometimes they can also be real numbers or can also be complex numbers. The complex numbers have an imaginary part attached to them. The flux of vector field is nothing but a flow property of carriers. This property is concerned with a unit area.

The flow is per unit area. This concept is very important in the field of physics also. Both mathematics and physics are related to each other. So, this concept appears in both physics and mathematics. The knowledge of physics will help in understanding of the concept. The mathematical plot vector field will explain the concept.

The pictorial representation of the concepts will help in the understanding of the concepts. So, the pictorial representations are always helpful.

The flow of a vector field is related to the flux of the field. These both concepts are interrelated and the understanding of one concept will help in the understanding of the other concept. The plot can be shown in the Cartesian plane using the ‘x’ and ‘y’ axes.

The vector field graph shows the orientation of the various carriers in the Cartesian plane. The carriers are shown aligned in the different directions. So, the orientation of the carriers can be easily understood with the help of this graph.

This graph shows the different directions associated with the carriers. The carriers are shown oriented in different directions and this can be understood from the graph. So, the graph can be very useful.

Introduction to Linear Algebra

Linear Algebra is one of the branches of Mathematics.  These helps with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are found in modern math thus, it is widely used in both abstract algebra and functional analysis.
It is also found in solving problems in analytic geometry.
 Its application is found extensively  the natural sciences and the social sciences.  This forms an intro to Linear Algebra.

 The mathematical approach of solving for variables in equations is Algebra.  
It does the manipulation of these variables using mathematical structures.
It is like taking algebra to another level by manipulating these variables using mathematical structures. This also helps to write the entire system of equations in the form of matrices. When expressed in the form of matrices this to study in depth of the individual parts of matrix which is known as vertex. A vertex is simply a representation of a coordinate.

Linear algebra is a branch of mathematics.  The creation of analytic geometry led to  the system of linear equations  acquire a new significance.
Linear-algebra is important for both pure mathematics and applied mathematics.    
Linear form of algebra when combined with calculus facilitates the solution of linear systems of differential equations.

Analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences   use the techniques of Linear-algebra.
Linear mathematical models are sometimes used to approximate Non-linear mathematical models because linear algebra is such a well-developed theory.
The real world applications need Linear-Algebra.  Algebra is all about linear system of equations and their solutions Linear-Algebra usually consists of the linear set of equations as well as their transformations on it.  Linear Equations  consists of  topics  like  Linear Equations , Matrices, Determinants, Complex numbers, Second degree equations, Eigen values ,Linear Transformations.

Linear algebra help to mathematics: It is a useful branch of mathematics.   Below listed is some of the applications of Linear form of Algebra.  Constructing curves, Least square approximation, traffic flow, Electrical circuits, Determinants, Graph theory, Cryptography.
It   is used to draw graphs. Most equations of linear algebra will   be a straight line. For example draw a graph for a vehicle travelling at a constant speed at various time intervals.
The graph helps in determining the    unknown variable that is the distance by plotting in on the graph.  This can also be used for   a multitude of different functions that is a ready tool for  lots of different real life functions.