The Linear Algebra now a days becomes a interdisciplinary subject. It helps to study directly the problem in Economics, Chemistry, Engineering, Architecture and Commerce also. Here we have used the power of Matrix Algebra to solve the network problem and to establish new form of K.V.L and K.C.L The form of Matrix Algebra gives the Euler expression of topology. Other applications of matrices are discussed here.

Here we are constructing the matrix from any arbitrary circuit and by different legitimate operation of Linear Algebra like row reduction method and others we are trying to apply in physical system .We will represent the matrix in such a way that automatically the representation of phenomena like , relations between voltage difference and current flow, Ohm’s Law, Kirchoff’s Current Law , Euler’s formula in Topology automatically results. Then we have tried to develop the actual matrix relations between nodes , loops , trees which is widely used in graph theory which is applicable in finding particular search like websites from a lot of information as a search engine. Here linear algebra is used to establish chemical balance and in other applications also.

A real-life problem involving linear algebra would have only one solution, or perhaps no solution. The purpose of this  article is to show how linear systems with many solutions can arise naturally. The applications here come from economics, chemistry,  physics , natural science and network flow[1]. The only equation Ax = b is the  form of  whole linear algebras. The  combination of  vector  Ax  can  be considered as a dependent or independent  combination of the rows or the  columns of matrix .  It is found that always direct solution or results are not available. But if we think geometrically then  we will reach almost to the actual solutions. The other form of matrix  solution using the inverse of A: x = A^-1b  has also new concept if we think geometrically again. Here we will show the  pictures in terms of combination of vector space [2] .  Without knowing the detailed interaction picture  only by structuring the problem in correct way  and using the proper steps of matrix operation  we can get the final results. These relations between input and output can be  replaced by  matrix  in spite of  polynomials. When linear algebra is used to describe the circuit connected with alternating sources (A.C) then the significance of complex number automatically results[3] .

Let us consider the flow of any physical quantity  by a number of different paths. The flow in general is controlled by nature of  material , nature of distribution of path and quality of path also.  This idea will help to construct the number of bridges such that maximum utility  can be obtained  or  design of city or any thing like this . Here we consider the flow of current  in various paths  and with  verity   of flow path . Here we will transform the phenomena in vector method  and finally by matrix construction .

The given direction indicates the flow of physical quantity in  absence of  real source. Picture of network and matrix formed  with the idea of node , edge , tree ,  loop and interrelationship in between them. The unknowns X_l, X₂, X₃, X₄ represent potentials or voltages at the nodes. Then AX gives the  variation of voltages across different  edges. Any flow is due to differences of  potentials. The nulls pace which is basically orthogonal to the  space produced by space formed by   rows  contains the solutions to  Ax = O.  Initially we consider that  all reference potentials  are zero as a result  all the potential differences are zero . This is with idea of boundary condition.  

Physically it means all   the variables ( x ) in the nullspace is a constant vector (c, c, c, c) that is basically grounding.

The null space vector is 

Let us consider the for a particular edge indicated by suffix j , the conductance can be expressed by C_{j ,} Resistance by 1/C_j   , current is Y_j and potential   with respect some reference level  by e_j . Then  Ohm’s law  can be expressed as   Y_j =c_je_j .This is same as E=IR  form. Now  we are attempting to write to find the potential drop  across any resistance of value R.  Let A is the matrix of directed flow graph then Ax indicates the potential differences between  all the nodes. Physically (-)Ax indicates the drops of potentials. If we use a  source of E.M.F  of energy b then  according to the similarity of Ohm’s Law the potential drop across the resistance  can be expressed   as e = b-Ax or y=C(B-Ax).  

 There is a more general  physical significance of  transpose of  matrix.

Next  we  write the matrix of the form    A^Ty=0 . Here  A^Ty=0 is exact form of  Kirchoff’s current law without source and A^Ty= f  form of  Kirchoff’s current law with source . Here  equation of motions satisfying the kirchoff’s law are

 Y₁-Y₃-Y₄=0 , Y₁-Y₂=0  , Y₂+Y₃-Y₅=0  and    Y₄+Y₅=0 .

A^Ty=0 is basically  a balanced equation and says about the conservation principle . Since no current  causes  no storage of  charge at any node.  Here  basis vectors are formed by nodes  of loop  construct the null space of  A^Ty .  These null space vectors are  indicated  by summation of  the complete the outer loop made by  Y₁,Y₂,Y₄ and Y₅. It is all about kirchoff’s law.

If we use the external source of  E.M.F =  f  then   A^Ty   = f.

Now by two equations :  Y =C(b-Ax)  or C^-1y  +Ax=  b

  And                                   A^TY  =   f  we construct the matrix in equilibrium  condition as

Topological View:

There is basically no Topological theory but the circuit theory  when viewed in terms of linear algebraic angle the  beautifully is found that the circuit theory follow a strong topological relationship.

 From  the picture we see  some edges are dependent and some are independent. Since independent edges are not in a position to construct loop  only , loops are produced by  dependent columns or edges. Here in our example with general theory 

The dimension of left null space  Dim  N(A^T) = m-r

Number of loops= Number of edges- (Number of nodes -1)

So  (Number of  Nodes)  - (Number of edges ) + Number of loops =1

This is Euler’s formula in the context that if we consider  nodes are of zero dimension like points ,edges are of one dimension which is basically line  and loops are of two dimension  having definite area. This idea deals with topological information which is used in network theory.

Balancing in chemical reactions

Chemical equations describe the quantities of substances consumed and produced by chemical reactions. For instance, when propane gas burns, the propane (C₃H₈) combines with oxygen (O₂) to form carbon dioxide (CO₂) and water (H₂O), according to an equation of the form .

 (x₁)C₃H₈  +(x₂)O₂=(x₃)CO₂  + (x₄)H2O .

Solution gives the coefficient or the actual proportion of chemicals for any particular reactions.

Application in economics 

 In any model of economics we have a large no of variables and these variables are related among themselves by various relations.  Now mathematical relationships are developed and arranged in linear combination form . Then we construct the matrix form and we in general made legitimate operation of matrices and  the operations are made according to the problem like demand ,supply , interrelations, correlations ,rogations and design demand.  In general we use the MATLAB program to handle such type of big data. Then we can clear conclusion    about the future of the  model . These mathematical modeling formation is giving a new window of  market research . Most of the modern Stock exchange use the big linear programming  .

In recent times , linear algebraic models are very important and has a remarkable growth in physical and social science. These models fulfill the demand of representation of actual system and we can go through the systems for our future need . A large number of linear equations construct linear algebra and this article is written with the aim that this old mathematics can be thought geometrically and the new angle of thinking helps us to investigate new relationship and formulas which are already present in different brunches. This concept is equivalent to a vector equations as well as matrix equations . These idea of geometry will play an essential role throughout many branches of physics , statics and mathematics . This is tried to explore here.

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2. Introduction to linear algebra , Fourth Edition Gilbert Strang , Cambridge Press

3. Linear Algebra ,Jin Ho Kwak , Sungpyo Hong .Birkhauser ,Berlin

4.Linear Algebra , G.Hadly ,Narosa Publishing House

5. Elements of Group Theory for Physicists , A,W ,Joshi , New Age International Publishers

6. Mathematical Methods for Physicists , G.B.Arfken ,H,W,Weber ,Academic Press

7. Applications on Unitary Matrices in Physical Systems , K. Sarkara ,Innovative Research Concept journal.