Adolescents and teenagers are often fascinated by the code languages used by the heroes of detectives, spies, and mystery heroes, be they Sherlock Holmes, Miss Marple, Jessica Fletcher, Ethan Hunt, or The Saint. Have you ever wondered how these detectives and spies decode these secret codes? The truth is, by using linear algebra. Linear algebra is a branch of mathematics that comprises vector spaces, countless measurements, and linear mapping between the spaces. Simply put, linear equation systems are employed for all types of investigations. If you are intrigued to learn more about the applications of linear algebra, read this blog.
What is Linear Algebra?
Linear Algebra is a branch of mathematics concerned with linear equations, linear maps, and their presentation in vector spaces with the help of matrices. Applications of linear algebra are essential to all areas of pure and applied mathematics. Apart from that, it is used in almost all fields of natural science and engineering. It describes many natural phenomena and allows for the examination of the efficiency of mathematical concepts and language.
What Are The Applications Of Linear Algebra?
Applications of linear algebra can be observed in the following areas:
1. Cryptography
It is the study of decoding and encoding undisclosed messages. With the help of electronic transactions and communications, reliable encryption techniques can be used. However, those techniques include modular arithmetic to decode or encode the messages. For example, the concept of matrix transformation is used for simple encoding methods.
The following example will help you understand the methods of encoding cryptography quite easily. To program a message, take the following steps:
Step 1: Allocate a specific number to every alphabetical letter to predetermine the short secret messages.
Step 2: Arrange the series of the numbers of each text in the square matrix form. Name the matrix A.
[Note: If the number assigned to the letters is lower than the number of the element, then fill the matrix with the zero elements.]
Step 3: To encode the message, first assume that B is a nonsingular square matrix. After that, multiply matrix A with matrix B.
Assume the matrix B as:
Therefore, B =
Let’s put the word BILA KOCKA” (a white cat) into a matrix to decode the secret message. In that case, you must change the word into matrix A and encode the text, and then you must apply the following:
Z = BA =
Now, multiply the matrix Z by B in the opposite form to decode the message:
= A
As matrix multiplication is not commutative, it must preserve the matrices with the precise product value. If you multiply the Z and B inverse matrices in reverse order, you will obtain:
=
Now, the secret message is “CERNY PSIK” (a variety of black dogs).
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2. Game theory
It is another important application of linear algebra. Game theory is a mathematical study that explains the number of probable options. The players formulate these options during the game. According to psychologists, the social communication theory is employed to reflect on the player’s opportunity against other players in the competition.
Although game theory applies to cards, board games, and other competitive games, it is also utilized in military planning.
Let’s understand the applications of linear algebra in game theory with an example.
- Rock, Paper, Scissors
It is one of the simplest examples of the Zero-sum game. A payoff matrix is used similarly to that of a Prisoner’s Dilemma’s payoff in the following ways:
Suppose you need to count the two players’ scores over several games. A point is added to the player with every win, and a point is subtracted with each loss. For a tie, the point is neither subtracted nor added to the score.
Then the payoff matrix will look as follows:
n = m = 3.
P1 = P2 = P3 = {Rock, Paper, Scissors}
P1 P2 =
This matrix borders on that of a skew-symmetric matrix. This indicates that the game is symmetrical.
If one player wins, the other one will lose. But if both players have a tie, then none of the players will get the point.
It is the basis of the zero-sum game. Here, every positive choice will bring about a result, while negative choices will bring about opposition.
3. Real Life
Linear algebra plays a vital role in the fields of Math, Science, and Engineering. It mainly concentrates on finding unknown quantities. Below are some of the linear algebra concepts that are used in practical life.
- Linear algebra is useful in checking the distribution of microwave energy in a microwave oven.
- It helps create ranking algorithms in search engines like Google, Yahoo, etc.
- Uses of linear algebra are common in recovering codes that have been corrupted during processing or transmission.
- Plays a key role in space studies.
- Linear maps are used to show a three-dimensional view of a two-dimensional plane.
- Used to examine the digital indications and encode or decode them. These can be signals from audio or video.
- Linear algebra maximizes the areas of linear programming.
- Utilized to verify the levels of energy in atoms.
4. Computer Science
Applications of Linear algebra are essential in the following areas of computer science:
- Recognition of patterns
- Graph theory, like social graphs
- Classification and Clustering of data
- Singular Value Decomposition for Suggestion Systems
- Graphics Programming.
- Various types of Artificial Intelligence (AI)
5. Engineering
Here are some of the applications of linear algebra in engineering:
- Find solutions through linear equations to practical engineering problems
- Utilize matrix operations to examine and design control systems
- Make use of least squares regression to correspond to experimental data and make forecasts
- Utilize linear transformations to operate images and signals in the digital signal distribution
- Apply linear algebra to create and examine communication networks
- Use linear algebra to maximize resource allocation and scheduling in project management
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6. Science
Linear algebra plays a significant role in the study of science, specifically in the areas of physics, mathematics, engineering, and computer science. Here are some of its uses in science:
- Linear algebra helps to resolve issues based on vibrations and sound, electrical networks, and optics.
- Employed for shaping the processes of elements in the universe and their interactions
- Linear algebra helps to investigate the connection between vector spaces and linear transformations.
- Modeling structures and examining the impacts of forces on structures.
- Designing computer graphics and machine learning algorithms.
7. Economics
With the help of linear algebra, you may scrutinize economics and other financial markets. It includes analyzing stock markets, risk management, portfolio optimization, and economic analysis.
8. Data Science
Linear algebra is an essential tool for examining large volumes of data. The following are its uses in data science:
- You may use linear algebra to develop and evaluate large datasets used in the fields of machine learning and artificial intelligence.
- Using linear algebra is also beneficial for computer vision, medical imaging, and signal processing.
9. Robotics
Linear algebra is an indispensable instrument in the following areas of robotics:
- For developing and manipulating robots
- Analyze problems in robotics, like those related to planning and direction of motion, and enhance robotic systems.
- Design autonomous systems, like self-driving cars and robots.
10. Geometry
The use of linear algebra is fundamental to geometry. Linear algebra is helpful in the contemporary presentation of geometry for defining fundamental objects like lines, planes, and rotations.
11. Functional analysis
The functional analysis examines the vector spaces. Linear algebra studies the components of vector spaces like matrices, functions, and polynomials and helps in performing their binary operations.
12. Studies of complex systems
Partial differential equations solve most physical phenomena. Linear functions use large matrices to control the parameters and find the solutions to partial differential equations very easily.
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A Few More Linear Algebra Applications
Find here, a few more applications of linear algebra
- Search Engine Ranking System: One of the most significant uses of linear algebra is in the development of Google. To construct the most sophisticated ranking algorithm, you may use linear algebra.
- Linear Programming: Optimization is a key application of linear algebra that linear programming more frequently uses.
- Prediction: You may use the linear models that linear algebra builds for predicting some objects.
- Signal Analysis: You may widely use linear algebra for encoding, analyzing, and manipulating signals such as audio, video, and pictures.
- Graphics: One significant aspect of graphics is projecting a three-dimensional scene onto a two-dimensional screen. The linear algebra explains this aspect that linear maps solely handle.
- Error Correcting Codes: Linear algebra plays a key role in coding theory. By using linear algebra, you may retrieve the encoded data that has certain modifications. Hamming code is an important error-correcting code.
- Facial Recognition: Principal component analysis is a type of automated facial recognition method that uses linear algebraic expressions.
The Bottom Line
By now, you will have gained a better understanding of the practical uses of linear algebra. Apart from solving linear equations, linear algebra plays a significant role in handling various other purposes. In case, it is challenging for you to solve linear algebra equations or if you would like to know more about the uses of linear algebra, approach us. We have skilled mathematicians as assignment helpers in our team. Using their expertise, our experts will help you with what you want and improve your subject knowledge.
