Math 220: Real-World Applications of Linear Algebra Report

This report explores the field of Linear Algebra, focusing on its real-world applications. It introduces Linear Algebra as a mathematical discipline dealing with linear functions, vector spaces, and matrices, highlighting its relevance in diverse areas such as search engine ranking, error-correcting codes, a...Read More

Running head: Math 220
Math 220
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Introduction:
1.
Linear Algebra is a field of mathematics that deals with linear functions and their
representations in vector spaces often with the aid of matrices. This has many real life
application some of which will be discussed in this report.
Some of the applications of linear algebra in our contemporary world is ranking search
engine, error correcting codes, graphics, signal analysis, linear programming, community
detection, quantum computing and others.
In this report the mathematic behind linear programming will be studied which uses a
method from linear algebra called the Gaussian elimination method. In fact the simplex
method in linear algebra is a just a sophisticated expansion of the Gaussian elimination
method.
The algorithm Gaussian elimination method is mainly used to solve systems of linear
equations. It is in reality a sequence of operations performed on the corresponding matrix of
coefficients. This method is often used to find the determinant of matrix to calculate the
inverse of an invertible matrix. This method was named after a German mathematician Carl
Friedrich Gauss who is thought to be the first person to have discovered the method.
2.
The range of application of this short but simple method is vast. Here we will briefly touch on
a few topics.
Gauss elimination method is used in enhancing image. The technique used for this is called
“Directional Gaussian Filter”. Image filtering such as fingerprint image and non-subsample
countour transform process is applied along with other techniques to attain good images from
thumb impressions.

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3.
Linear programming is a part of an optimization problem that is used to solve a wide
range of practical problems. Optimization is where computer science and mathematics are
used together to solve real world problems. Common optimization problems are when,
factories want to maximize efficiency, organizations wanting to maximize profit. The tools
of optimization are a whole another world and an entire field of study called Optimization
Research is devoted to exploring further in the topic.
The simplest of the optimization problem is the linear program and Gaussian
elimination plays a very important role in it. Optimization problems generally have two parts:
an objective function, the thing we want to minimize or maximize, and the constraints i.e. the
rules that need to follow while maximizing or minimizing the objective function. For
example, one may want to find the shortest route to go somewhere (objective function) but
cannot possibly have negative or zero distance to cover in any possible way (constraints).
A large and diverse variety of problems can be modelled as linear programs. Some such
examples are:
Maximizing an outcome: Deciding where to drill for a mineral, given a set of other relevant
constraints.
Scheduling Tasks: for example when scheduling flight crews for airlines.
One of the most commonly used linear programming methods is the simplex method. It is
available in most computer packages. The algorithm is known for its reliability and working
over the years have shown that Simplex has repeatedly good performance in practice on real
problems.
The Simplex method works in a way similar to the Gaussian Elimination method as it is
derived from the later.

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For illustration, simple is solved below:
X1 + 3X2 – 4X3 = 8
X1 + X2 – 2X3 = 2
−X1 – 2X2 + 5X3 = −1
The idea is to solve the system of equations using strategic matrix operations.
The following methods can be applied to the matrix or the equations to help solve the system.
1. Multiplying the equations by any constant
2. Adding or Subtracting equations
3. Interchanging equations
Gaussian Elimination is systematic method of applying the above techniques to make the
value of one variable obvious and then find the values of other variables by back substitution.
The basic steps and idea for the Gaussian elimination method are:
1. The first variable from except the first equation can be eliminated by adding a suitable
multiple of the first equation to each of the remaining equations
2. The second variable from except the first two equation can be eliminated by adding a
suitable multiple of the second equation to each of the remaining equations
3. And so on ….

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Taking a simple linear programming problem can shed light how this process is helpful and
can be solved easily by the Gaussian elimination method.
A common problem is that one the distribution of diet that sustains a healthy life. Suppose, as
recommended by WHO, people need 3.7 litres of water, 1000 mg of Ca per day, and 90 mg
of vitamin c per day.
If it is needed to attain that level of nutrition by buying oranges, whole milk and broccoli, the
problem that is needed to solve is what combination of the above three will be able to attain
the required nutrition in the minimum cost.
The cost of the food can be expressed mathematically as: 0.3b + 0.1m + 0.2r.
The constraints can be written as
which can be modelled as a matrix. Thus our problem reduces to solving the matrix to
minimize the cost function. The matrix can be solved methodically by the simplex method
which is derived from the Gaussian elimination method.

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References:
Hogben, L. (2013). Handbook of linear algebra. Chapman and Hall/CRC.
Robert, A. M. (2015). Linear algebra: examples and applications. World Scientific
Publishing Company.