What Are The Top Applications Of Linear Algebra?

Applications Of Linear Algebra

A spy detective game is one of my favourite games in my childhood. And I think you are not an exception who never play this game. In my childhood age, I always imagined how detectives decode secret messages? And I was always curious to know about it. 

When I grow up, I am able to find out the method to decode the message. What is that? Well, it is one of the applications of linear algebra called Cryptography. Below, I have explained the exciting applications of linear algebra in detail. So, let’s move to the details of one more concept of mathematics. 

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What is Linear Algebra?

It is a branch of mathematics that includes vector spaces, infinite dimensions, and linear mapping among the spaces—the system of linear equations used for this kind of investigation. 

The linear algebra equations are represented with the help of the formalism of vectors and matrices. Moreover, linear algebra is necessary for both applied and pure mathematics. 

Key Point:
Linear algebra relates to linear combinations. It means you can use arithmetic on the numbers’ column that is known as arrays and vectors of the numbers termed as matrices. These matrices are created for new arrays and columns of the numbers. 

Let’s take an example of it; the abstract algebra method is used for relaxing the vector space’s axioms. The functional analysis use for the infinite-dimensional vector space’s version. 

On the other hand, linear algebra helps in providing the linear systems’ solution for differential equations. 

There are several techniques of linear algebra that are used for physics, computer science, analytic geometry, natural science, computer animation, and social science (particularly in economics). 

Being a well-developed theory, linear algebra is used for non-linear mathematics models approximated by linear details. 

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What Are The Applications Of Linear Algebra?

Here are some of the application of linear algebra:

1. Cryptography

It is the study of decoding and encoding of the secret messages. Using electronic transactions and communications, solid encryption methods can be applied. Those methods involve modular arithmetic to decode/encode the messages. And the simpler encoding methods apply using the concept of matrix transformation. 

I mentioned above that the linear algebra concept is used to decode the secret message (a cryptography method). 

Let’s understand it with an example.

Assign the specific number to each alphabetical letter to encode the short secret messages. 

Then, the sequence of the number of each text should be organized in the square matrix form (taken as A). 

[Note: if the letters’ number is lower than the number of the element, then fill the matrix with the zero elements.]

Assume a nonsingular square matrix as B. To encode the message, multiply matrix A with matrix B (i.e., matrix A * matrix B). Assume the matrix B as:

B = 

Here is the text ”BILA KOCKA” (a white cat) change into matrix A: 

and to encode the text, apply: 

Z = BA =

Now, multiply the matrix Z to B inverse to decode the message:

As matrix multiplication is not commutative, it must maintain the matrices with the exact product value. If you multiply the Z and B inverse matrix in the opposite order, you will obtain:

Now, the secret message was “CERNY PSIK” (a variety of black dogs).

2. Game theory

It is another one of the applications of linear algebra, which is a mathematical study that describes the number of possible options. The players make these options during the game playing. As per psychologists, the social interaction theory is used to consider the player’s options against other players in the competition. 

Although game theory focuses on cards, board games, and other competitive games, it also applies to the military strategy used in wars.

Let’s understand it with an example

Rock, Paper, Scissors 

It is one of the simple examples of the Zero-sum game. A payoff matrix is used similarly to that of Prisoner’s Dilemma’s payoff. 

How?

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 substracted nor added to the score. 

Then the payoff matrix will look as:

This matrix is similar to that of a skew-symmetric matrix. This implies that the game is symmetrical.

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This matrix is similar to that of a skew-symmetric matrix. This implies that the game is symmetrical. 

If a player wins, the other one will lose. If both tie, then the players will not get and detect the point.

It is what the zero-sum game is. 

[Each positive option will result in a reward, and a negative option will lead to opposition.]

Are there any other applications of linear algebra?

Yes, there is!

Besides the above-mentioned applications of linear algebra, the concept is also used in:

1. Networks and graphs for analyzing networks.
2. Matrices in the field of engineering, like a springs line.
3. The simplex optimization method for linear programming.
4. Linear algebra for probability and statistics like least squares for regression.
5. Fourier Series is used in signal processing using various functions.
6. Chemistry and physics for equilibrium temperature distributions.
7. Leontief’s economics models describe the interrelations between outputs, prices, and demands in the economic systems.
8. Computed tomography that uses for constructing a human body’s cross-sectional view using X-ray scans.
9. Genetics looks at the concept behind dominant and recessive traits.
10. Anything that includes data fitting or differential equations.
An interesting thing about linear algebra:- 
Linear algebra is the type of mathematics that is also used by the famous scientist Albert Einstein. He used it in the theory of relativity, particularly tensor calculus and tensors. Moreover, he introduced the concepts of linear algebra notations to physics. Those notations are called Einstein summation convention or Einstein notation.

What Are The Five Applications Of Linear Algebra?

Here in this section we will tell you the five application of linear algebra:

Application 1Making Curves and surfaces passing through Specified points
Application 2Least Square approximation
Application 3Traffic Flow
Application 4Electrical Circuits
Application 5Determinant

Application of Linear Algebra in Real Life: Where it is Used? 

Linear algebra is widely used in the fields of Math, Science, and Engineering. Basically, it plays a vital role in determining unknown quantities. Below are some of the linear algebra concepts that are used in real life.

  • Linear Algebra is used to check the distribution of microwave energy in a microwave oven.
  • It is used to create ranking algorithms in search engines such as Google, Yahoo, etc.
  • Used to recover the codes that have been tampered with during processing or transmission.
  • Used for space studies.
  • It is used for projecting a three-dimensional view into a two-dimensional plane, handled by linear maps.
  • Used to examine the digital signals and encode or decode them. These can be the signals of audio or video.
  • It is used to optimize in the field of linear programming.
  • Used to check the energy levels of atoms.
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Suppose you are interested in computer science and want to know where linear algebra applications are used in computer science. We listed six linear algebra applications that are used in computer science.

Applications of Linear Algebra in Computer Science

Linear algebra is essential for things like:

  • Pattern Recognition.
  • Graph theory (social graphs, for example).
  • Data Classification and Clustering.
  • Singular Value Decomposition for recommendation systems.
  • Graphics Programming.
  • Various forms of Artificial Intelligence (AI).

Applications Of Linear Algebra In Engineering

Here are some of the applications of linear algebra in engineering:

  • Solving systems of linear equations to model real-world engineering problems.
  • Applying matrix operations to analyze and design control systems.
  • Employing least squares regression to fit experimental data and make predictions.
  • Utilizing linear transformations to manipulate images and signals in digital signal processing.
  • Applying linear algebra to design and analyze communication networks.
  • Using linear algebra to optimize resource allocation and scheduling in project management.

Applications of Linear Algebra in Economics and Finance

Linear algebra plays an essential role in the realms of economics and finance by providing powerful tools to analyze and model complex systems. Within these fields, it aids in optimizing portfolios, understanding economic relationships, and developing sophisticated models for decision-making processes.

  • Portfolio Optimization: Linear algebra facilitates the creation of efficient portfolios by employing techniques such as Markowitz’s mean-variance optimization, using matrix operations to find the optimal allocation of assets that balances risk and return.
  • Input-Output Analysis: It enables economists to model interdependencies between different sectors of an economy using input-output tables, which are essentially matrices, to understand the ripple effects of changes in one sector on others.
  • Econometrics Modeling: Linear regression and multivariate regression models in econometrics heavily rely on matrix operations, allowing economists to estimate relationships between multiple variables, forecast trends, and test hypotheses efficiently.
  • Markov Chains in Economic Models: Utilizing Markov chains, which are modeled through matrices, helps economists analyze dynamic processes in economic systems, such as forecasting economic transitions and studying state-dependent events.
  • Optimization Problems in Economics: Applications for linear programming include production scheduling, resource allocation, and economic decision-making. It is based on the optimization of a linear objective function under linear constraints.

Applications Of Linear Algebra In Machine Learning

Linear algebra serves as a foundational pillar in the realm of machine learning, providing essential tools and techniques for data manipulation, dimensionality reduction, and model optimization. Its applications range from basic linear regression to complex neural network architectures, enabling the handling of vast datasets and the creation of sophisticated learning algorithms.

  • Linear Regression: Utilizes matrix operations to fit a linear model to data, determining relationships between input variables and the target variable.
  • Principal Component Analysis (PCA): Employs matrix decomposition techniques to reduce the dimensionality of data, identifying the most important features and aiding in visualization.
  • Neural Networks: Utilizes linear algebra for computations in layers of neurons, including matrix multiplication in feedforward and backpropagation algorithms, enabling complex pattern recognition and decision-making.
  • Singular Value Decomposition (SVD): Helps in matrix factorization for recommendation systems and image compression, enabling efficient representation and reconstruction of data.
  • Optimization Algorithms: Gradient descent, a fundamental optimization algorithm in machine learning, heavily relies on linear algebraic operations to update model parameters iteratively for better accuracy and convergence.

Conclusion

Linear algebra basically is the study of the planes and lines, mapping, and vector spaces, which are needed for linear transformations. Therefore, it is necessary to know what are the applications of linear algebra. Above I have explained those applications. Understand them deeply and try to use the concept of linear algebra in real life. 

In case you are not able to solve or understand the concept of linear algebra, comment in the below section. I will help you in the best possible way that will definitely be going to help you in the long run. and also get the algebra homework help and math algebra homework help from our experts. 

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Frequently Asked Questions

Q1. Where is linear algebra used in real life?

Linear algebra can be utilized nearly in all compute-intensive work and tasks. It can also be used efficiently to solve any nonlinear or linear equations.

Q2. How is linear algebra used in statistics?

Linear algebra use in any application, which deals with multiple random variables at a particular time. But it has been seen that linear algebra is mostly used in multivariable statistics course(s).

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