This article is very special to me, As I am involved in Computer Science (since 2010) and Recently Started Learning Linear Algebra (since 2020), And I love em both.
This article will discuss how computer science uses underlying concepts of linear algebra to solve complex problems.
Linear Algebra is a branch of mathematics that deals with matrices and vectors. Matrix and vectors are special data structures that are used to store, represent and manipulate data.
The common application of Linear Algebra in CS includes image processing, signal processing, Network analysis, Machine Learning and many more.
Linear Algebra is backbone of computer science. Major tasks like Image Processing, Signal Processing, Network Analysis, Computer Graphics, Cryptography, and many more uses linear algebra to solve complex problems.
Image is collection of pixels. Collection of pixels is nothing but a matrix. Each pixel is a value that represents the intensity of the color. The color of the pixel is represented by the combination of Red, Green and Blue (RGB) values. A computer cannot save image as it is, it saves it as a matrix of pixels.
Linear Algebra operations are used to manipulate these matrices to perform operations like scaling, rotation, applying filters, etc.
Applying filters is just a act of multiplying the image matrix with a filter matrix. Sounds so simple, right? Its get more simpler when you start to understand the concepts of linear algebra.
Signal Processing is the method of analyzing, modifying and synthesizing signals like sound, image, etc. Signals are nothing but a collection of data points. These data points are stored in a matrix. Linear Algebra operations are used to manipulate these matrices to perform operations like filtering, noise reduction, etc.
Network is simply a collection of nodes and edges. But how these nodes and edges are interpreted by a computer? How does a computer understand the network? The answer is simple, using matrices.
In computer networks, understanding how nodes, servers and computers are connected is crucial.
Matrices, specifically adjacency matrices and laplacian matrices are used to represent the network. These matrices are used by computer to keep tract of connection, understand the network path and many more.
Machine Learning is all about huge data and algorithms. There is a lot of data that needs to be fed into algorithm for 100s of thousands of iterations.
To manage the structure of this data, matrices are used. And Matrices have special operations like multiplication, finding determinants, addition and inverse that are used to manipulate the data. It becomes very easy to work with data in matrix form.
Read More about Linear Algebra in Machine Learning
The images you see on your computer screen are nothing but a collection of pixels, to be more specific, a matrix of pixels.
Each pixels is represented by a value that represents the intensity of the color. The color of pixel is represented by the combination of Red, Green and Blue (RGB) values.
Matrices are used to represent these pixels and linear algebra operations like multiplication, transpose, etc are used to manipulate these matrices of pixels.
are nothing but matrix operations that are used on the matrix of pixels
Cryptography is a way of securing information (data) so that only people or devices who are intented to read and access the data can understand it.
Linear algebra is widely used in algorithms that are used to encrypt and decrypt the data. Although each algorithm have their own way of encryption/decryption operation.
However, most of them use matrices to represents the data in bits and bytes and then perform operations on these matrices to encrypt and decrypt the data.
For example, I have a matrix A which represents my data.
\(
A = \begin{bmatrix}
1 \\
2 \\
3 \\
4
\end{bmatrix}
\)
and I have a key matrix K (which is used to encrypt the data)
\(K = \begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1
\end{bmatrix}
\)
To encrypt the data, I multiply the data matrix with the key matrix.
\(A \cdot K = \begin{bmatrix}
1 & 2 & 3 & 4
\end{bmatrix} \cdot \begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1
\end{bmatrix}
\)
The result of this multiplication is the encrypted data.
This is just a simple example, there are many complex algorithms that are used in cryptography.