This document provides a comprehensive overview of linear algebra, including its definitions, key concepts, and applications in finance. It discusses the importance of linear equations, matrices, and vectors, and highlights how these mathematical tools can be used to solve real-world problems, particularly in financial contexts. The document also addresses different types of linear equations and their behaviors, supported by examples and theorems.

1. Linear Algebra and its use in finance:
Introduction
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds
of solutions which are related to linear equations. In order to explain the Linear Algebra, it is
important to explain that the title consists of two different terms. The very first term which is
important to be considered in the same, is Linear. Linear may be defined as something which is
straight. Linear equations can be used for the calculation of the equation in a xy plane where the
straight lines has been defined. In addition to this, linear equations can be used to define
something which is straight in a three dimensional perspective. Another view of linear equations
may be defined as flatness which recognizes the set of points which can be used for giving the
description related to the equations which are in a very simple forms. These are the equations
which involves the addition and multiplication.
The process of learning of linear algebra is tough and complicated. But most of the part of
algebra is concentration towards the objects which are either straight or flat. In doing so, there is
not a requirement to visualize the same.
In this paper, the key concepts which are related to linear algebra which includes the use of
equations, matrices and vectors will be discussed, the theorems related to the same will be
analyzed. After this, the important of Linear algebra will be understood in finance. This will help
to develop a sense of information towards linear algebra.
Application of Linear Algebra

2. Linear algebra forms to be one of the most important part of mathematics. It is a branch which
has a large number of applications in the real world. Because of its ease in solving, it can be used
in different parts of science which the approximation of equations is done with the help of linear
equations. Linear algebra exists in different forms such as abstract algebra, vector spaces and
matrics and have been used for different kinds of theories of mathematics such as the group
theory, ring theory etc (Noble, Ben, 1988). By the process of understanding of different kind of
tools and theorems which are related to linear algebra, it is important to realize the importance of
the functional implementation of the same. One example where numerical application has been
widely used is the Numerical processing of images. This is the branch of algebra where, the
algorithm helps to select the usual information for the discretion of the users. They are also
useful in the linear control theory. The system can be directly described as the part of the state
which is the vector and the change of state is identified by the matrices.
Solutions to linear equations
In order to solve the different kinds of linear equation one needs to understand the forms of
linear equations which has been existing. Suppose, one assumes the following sets of equations:
x2
+ y2
= 1
-x + √3y = 0
In order to solve the equations, the substitution has been done in such a manner that the possible
values of x = √3/2, y=1/2 and x=-√3/2 and y=-1/2. After this one needs to find the plot of the
solutions in the xy plane. In order to do this, a circle is taken whose center falls to be the origin
with a radius of 1 and draw the straight line through the same with the slope 1/√3 in order to get
the equations true(Beezer & Robert Arnold, 2008). After this has been obtained, the desired
solutions will be obtained. The solutions of linear equations can be written assets:

3. S: { (√3/2, ½), (-√3/2, -1/2)}
Definition
A system of linear equation may be defined as the collectionof different kinds of equation in
which the variable is the quantities x1, x2,x3, x4 …xn of the form:
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3
am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
A solution of the system of linear equations may be defined as the set which consists of each of
the solution in the equation and nothing else.
For example:
X1+ 2x2 + x4 = 7
x1 + x2 + x3 -x4 = 3
3x1 + x2 + 5x3 - 7x4 = 1
In the above set of equations, there are n=4 variables and m=3 variables.
The possibilities of the set of solutions in a linear equations can be explained with the following
examples:
2x1 + 3x2 = 3
x1- x2 = 4
While plotting the solutions to the same on an x-y plane one will be getting two lines. While one
of them will be having a positive slope, other will have a negative slope. But they will be having
a common point which is (x1, x2=(-3, 1) which will also be the solution to the equation. Thus,

4. according to the geometry it can be believed that it will be the only possible solution, and it is
unique.
2x1 + 3x2 = 3
4x1+6 x2 = 6
If the plot of these equations is draws, one line will be on top of another. Thus, there will be
many points which will make the equation true.
Another examples can change the possible sets to parallel lines:
2x1 + 3x2 = 3
4x1+6 x2 = 10
This equation will be giving the identical slope or the parallel lines.
Thus, the set of linear equations can possess different kinds of behaviors. It can either be parallel,
meet at a single point or meet at infinite points.
Equivalent system of linear equations:
The two linear equations are equivalent if their solution sets are equal. Let us assume that there is
a set of two linear equations. The following operations will transform them in a different one.
This operation will be called as equation operation.
Theorem 1: Equation operations preserves the solution sets:
According to this theorem, if one of the operations is applied to the system of linear equations,
then the original system and the transformed system can be equivalent.
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3

5. am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
It is assumed that S denotes the solution to the statement of the theorem and T denotes the
solution to the system.
a) S is a subet of T. If (x1, x2, x3….xn)= β1, β, β3… βn € S is a solution for the original
system of equation. If we ignore the i-th equation, it can be said that the other equation of
the transformed system are true:
aαi1 x1+ a α i2 x2 a α i3 x3 …..a α in xn = α βn
This states that the i-th equation of the transformed system is also true, thus β is also a subset of
the equation.
Theorem 2: Three equations one solution, can be solved by done by the following sequence of
equations:
x1 + 2x 2 +2x3 = 4
x1 + 3x 2 +3x3 = 5
2x1 + 6x 2 +5x3 = 6
Now in the above equations, a=-1 times equation is done to get the following solution:
x1 + 2x 2 +2x3 = 4
0x1 + 1x 2 +1x3 = 1
2x1 + 6x 2 +5x3 = 6
Now a=-2 times the equation 2, added to the equation to get the result:

6. x1 + 2x 2 +2x1 = 4
0x1 + 1x 2 +1x3 = 1
0x1 + 2x 2 +1x3 = -2
From the above results, it can be obtained that
x1 + 2x 2 +2x3 = 4
1x 2 +1x3 = 1
x3=4
Theorem: In order to denote the columns of the m*n matrix, A having the vectors as A1, A2, A3-
--An. Thus, x may be defined as the solution to the linear system of equations LS (A, b) if and
only if b is equal to the combinations of the columns related to A which are formed with the
entries of x1.
[x]1 1 A1 + [x] 2 A2 + [x] 3 A3 + ··· + [x]n An = b
Suppose, if there exist an entry into the coefficient matrix A which is related to the row I and the
columns j which has two names aij can be defined as the coefficient of xj.
Matrix and Vector System of Equations:
A m*n matrix may be defined as the rectangular layout of numbers from C having m rows and n
columns. The rows of the matrix can be reference to start at the top and to work down (i.e. row1
is at the top) and the columns will be referenced to start from the left (column 1 is at the left).
The matrix is noted as Ai and the notation is written in the form of: [A]ij.An example of the
matrix with m=3 rows and n=4 columns can be defined as :
-1 2 5 3
1 0 -6 1
-4 2 2 -2
Augmented Matrix:

7. In order to explain the augmented matrix, it is important to consider the system which has m
equations present in n variables. Each of these have the coefficient matrix A and a vector
constraint as b. In such a case the augmented matrix of the system is the matrix in which
m*(n+1) matrix is the one whose first n columns are the one who is the part of Matrix A and the
last column of the same is the column vector b.
The augmented matrix can be written as: [A | b].
Row operations:
Row operation can be performed by the following. This means the swapping of two of the
locations, multiply each entry related to the single row with a non-zero quantity and multiplying
the values to the entries which are there in the same column in the second row.
Theorem:
Row equivalent matrices
Vector
Vectors can be of different types and can include column vectors ordered list consisting of m
numbers, which are written in a vertican order. This starts from the top and proceeds to the
bottom. The column vector can also be referenced as a vectors. The column vectors are noted by
writing the same in bold using smaller alphabets.

8. Definition of Zero column vector:
A zero column vector may be defined as the vector of size m, where each and every entry of the
vector is 0.
0
0
0
Vector of constraints:
For a given system of equations,
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3
am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
vector of constants may be defined as the column vector of size m
b1
b= b2
bn

9. Solution Vector:
Solution vector may be defined as the column vector of size n :
x1
x2
xn
Theorem of Vector Space Properties of column vectors:
It is assumed that the Cm
may be defined as the set of column vectors having the size m with
addition and multiplication scales, then:
• ACC Additive Closure, Column Vectors
if u, v € Cm
, then u + v € Cm
• SCC Scalar Closure, Column Vectors
If α€ C and u€ Cm
then αu € Cm
• CC Commutativity, Column Vectors
If u, v € Cm
then u+v=v+u
• AAC Additive Associativity, Column Vectors
If u, v, w € Cm
, then u+(v+w)= (u+v)+w.
• ZC Zero Vector, Column Vectors
There is also a vector 0 which is known as the 0 vector, such as the u+0=u, for all u Cm
• DSAC Distributivity across Scalar Addition, Column Vectors
If α, β € C and u € Cm
, (α+ β)u= αu + βu
• OC One, Column Vectors
If u € Cm
, u=v.
• AIC Additive Inverses, Column Vectors
If u € Cm
, then there is an existence of a vector where –u € Cm
, so that u+ (-u)=0.
Proof: Some of the properties are the one are obvious, bt it is important that the proof can be
given for them. Though prove for the same is quite tough.

10. For DSAC, the proof can be given as:
[(α+ β)u]i= (α+ β) [u] i
= [αu] ] i + [βu] i
Because of some of the components of the vectors (α+ β) are equal and αu + bu are equal for all,
1<i<m, proves that the vectors are equal.
Theorem Vector Form of Solutions to the Linear equations:
Let us assume that [A | b], may be defined as the augmented matrix which consists of the system
of linear system which is consistent LS(A, b) of m different kinds of equations which are present
in n different variables. Let us assume that B is a row-equivalent of m* (n+1) matrix and it is
reduced in the row-echelon form. Also, if B has r non zero rows, columns without leading 1’s
with the indices and the columns which are leading 1’s having indices D= {d1, d2, d3, ….dr). The
vectors can be defined as:
[c]ij= 0 if i€F
[B] k, fj , if i€D, i=dk
[Uj]I = 1 if i € F, i=fj
0 if i € F, I != fj
-[B]k, fi if i€ D, i=dk
Use of Linear equation in finance:
Linear equation are used in a large number of financial equations and can be used to study a
large number of financial context. Linear algebra can be used for the purpose of representing the
potential schemes related to pricing and can help to compare different models at different points
or price points. There are a number of financial situations where linear equations sort to be very
useful. With the help of the system of calculation of one variable, the other variables can be
easily known.

11. It is used in finance, for a large number of purposes such as to give the amount of interest which
may be accrued on a line of credit after given point of time. In order to explain the example of
the use of linear equation in financial context, a situation can be considered where one needs to
have money to make improvements in the credit line in a given point of time (Berry et al, 1995.
By consideration of the situation and doing the calculations using the linear equations, one can
assume the amount which has to be paid with an interest rate of one year. One can decide to pay
off in a go or through the easy monthly installments as per his choice. In each and every process
of banking, linear equations can be used for the purpose of getting the rate of interest, the total
amount and the number of years which would be required to pay the amount. The complete
banking process is done with the help of linear equations can be implemented with the help of
linear equations.
References
Noble, Ben, and James W. Daniel. Applied linear algebra. Vol. 3. New Jersey: Prentice-Hall, 1988.
Beezer, Robert Arnold. A first course in linear algebra. Beezer, 2008.
Berry, Michael W., Susan T. Dumais, and Gavin W. O'Brien. "Using linear algebra for intelligent
information retrieval." SIAM review 37.4 (1995): 573-595.