Introduction to Linear Algebra
The study of Economic models reduces to the study of multiple equations whose simplest form
is Linear Equations.
Linear Systems
A Linear system is called linear because its representation in Euclidian plane is a straight line.
Its is of the form
𝑎1𝑥1 + 𝑎2𝑥2 + ...... + 𝑎𝑛𝑥𝑛 = 𝑏 where a1, a2, an are called parameters ( constants ) and x1,
x2, xn are called variables.
We study linear algebra because it transforms the quotations into simpler understandable
insights into planar geometry, moreover :
● Linear systems help in calculating exact solutions to the equations
● Linear systems help determine the precise relationship between the solution of the
equation and various parameters.
Although the world around us is generally non-linear, we can still get a good approximation of it
using a linear system. We use the same in calculus, which uses the tangent and slope at a point
in linear form to get a linear approach.
Using linear equations we find whether the function is increasing/decreasing and also its
maxima/minima
SYSTEMS OF LINEAR EQUATIONS
An Implicit Linear equation system is one where we have the exogenous and endogenous
variables mixed up with each other in same side of equal sign.
Example
2𝑥1 + 3𝑥2 = 7
5𝑥1 + 4𝑥2 = 9
GAUSSIAN AND GAUSS-JORDAN ELIMINATION
Lets consider a system of 3 linear equations of 3 unknown:
10𝑥1 − 4𝑥2 − 3𝑥3 = 130
− 2𝑥1 + 8. 8𝑥2 − 1. 4𝑥3 = 74
− 5𝑥1 − 2𝑥2 + 9. 5𝑥3 = 95 —---------------- (1 )
We denote this system by (1)
Now Gaussian Elimination tells us to eliminate one variable each until we get a system of linear
equations where every subsequent equation has one less variable than the previous one and
the last equation has only one.
For this we should
(1) Multiply the both side of the equation by a non-zero scalar and add/subtract with the
subsequent equation
For example :2𝑥1 + 3𝑥2 = 7
5𝑥1 + 4𝑥2 = 9, here we multiply first equation by -5/2 both side we get
− 5𝑥1 − 15𝑥2/2 =− 35/2 , now we add this new equation to 2nd equation from original system
we get − 7𝑥2/2 =− 17/2
(2) Interchange two equations if the subsequent equation don’t have the variable to do the
multiplication and addition for elimination
Now start this process in our original (1 ) system.
We multiply (1a) by 1/5 onboth side we get 2𝑥1 − 0. 8𝑥2 − 0. 6𝑥3 = 26
We add this transformed equation with (1b) we get
8𝑥2 − 2𝑥3 = 100 , we will use this as part b of a new system using part a and part b of original
system
Similarly we multiply (1a) by 0.5 on both side we get 5𝑥1 − 2𝑥2 − 1. 5𝑥3 = 65 , adding this
with(1c) we get − 4𝑥2 + 8𝑥3 = 160, we will use this as part c of a new system using part a of
original system
10𝑥1 − 4𝑥2 − 3𝑥3 = 130
8𝑥2 − 2𝑥3 = 100
− 4𝑥2 + 8𝑥3 = 160 —---------------- (2 )
Clearly as (2b) and (2c) are modification of (1b) and (1c), this system will have the same
solution as (1)
Now we will multiply (2b) by 0.5 on both side, we will get 4𝑥2 − 𝑥3 = 50 and we will add this with
(2c) to get 7𝑥3 = 210, we will use this a part c of a new system using part a and b of system (2).
The new system will be
10𝑥1 − 4𝑥2 − 3𝑥3 = 130
8𝑥2 − 2𝑥3 = 100
7𝑥3 = 210 —---------------- (3 )
Clearly this system if equation (3) will have same solutions as (3b) is just modification of (1a)
and (1b), (3c) is modification of (2b) and (2c) ( system 2 also have same solutions).
IMPORTANT : WHEN WE USE ELEMENTARY OPERATIONS LIKE
MULTIPLICATION/DIVISION/ADDITION/SUBSTRACTION ON BOTH SIDE OF EQUATIONS,
THE SOLUTIONS REMAINS UNCHANGED
What we have achieved in system (3) is the first part of Guassian Elimination where each
susequent equation has one less variable
Now for the second part of Guassian Elimination we will use the method called Back
Substitution.
● Where we start with equation (3C) which is single variable problem and will solve for
𝑥3 = 30,
● We will use the value of 𝑥3 = 30 in equation(3b) which will simplify to 8𝑥2 − 60 = 100,
which give 𝑥2 = 20
● Now we will use the value of 𝑥3 = 30 and 𝑥2 = 200 in equation (3b), which will simplify
to 10𝑥1 − 80 − 90 = 130 which will give 𝑥1 = 30.
TO summarize Guassian Elimination tells us to
1. eliminate one variable each until we get a system of linear equations where every
subsequent equation has one less variable than the previous one and the last equation
has only one.
2. Back substitute the value of variables obtained in the equation with one variable to its
previous equation to get the value of its previous variable until we get every value.
ELEMENTARY ROW OPERATIONS
As we saw in previous equations the elementary operations we used in different equations of a
system is concerned with coefficient only. For example :
𝑎11𝑥1 + 𝑎12𝑥2 +−−−−+ 𝑎1𝑛𝑥𝑛 = 𝑏1
𝑎21𝑥2 + 𝑎22𝑥2 +−−−−+ 𝑎2𝑛𝑥𝑛 = 𝑏2
- - - - - - - -
- - - - - - - -
𝑎𝑚1𝑥2 + 𝑎𝑚2𝑥2 +−−−−+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚
Here what we are concerned with is 𝑎1 and 𝑏1 not with x1and x2
Thus we denote these coefficients into rectangular arrays like
Here A is called Coefficient matrix
We can also add an extra column with the coefficients b1, b2,.... bn
This is called Augmented Matrix
Now for linear equation systems we performed some elementary row operations like to
achieve simpler result using Guassian Elimination which were
1. Interchanging the order of two equations of a system
2. Transforming an equation by adding a scalar multiple of other equation.
3. Multiplying an equation by a non zero scalar no on both side
We know that the solution to the system doesn’t change by these as these are elementary
operations.
We use the same three operations in matrix operation as it will give equivalent matrices with the
same solution, we will
1. Interchange any two rows of the matrix.
2. Change a row by adding a multiple of any other row to every element
3. Multiplying each element of the row by the same non-zero number.
These are Elementary row operations on a matrix performed to reduce matrix to simpler matrix
which gives solution easily.
Like for equation (1 )
