 # What Are The Top Applications Of Linear Algebra?

A spy detective game is one of my favorite 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.

## 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.

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.

## What are the applications of linear algebra?

### 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:

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

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and to encode the text, apply:

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).

### 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 the 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 the 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 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.

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

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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:

## 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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