Linear Algebra

Introduction to Linear Algebra

Let take a look at an introduction to linear algebra. As the title may suggest, linear algebra is a branch of mathematics. It is concerned with mathematical structures. It deals with mathematical structures that are closed under the operations of addition and scalar multiplications. It also includes the theory of linear equations, matrices, determinants, vectors, spaces, and linear transformations. It is essential to understand the concept of linear algebra. It is the principle theory behind the understanding of Machine learning. Linear algebra gives you better grasping power for how algorithms work. This knowledge also helps you to make better decisions. So, it advisable to master this concept as it gives you an insight as to how linear algebra is used for Machine Learning.

Linear Algebra - The Basics

Linear algebra is a continuous form of mathematics. It has applications even in the science and engineering streams. It allows you to compute with efficiency. It is that branch of studies that is central to Mathematics. It is valuable in geometry, statistics, and functional analysis.

Linear equations represent the data. Now, this data is presented in the form of vectors and matrices. As we all know, that data in statistics is always about numbers. Numbers are scalars. Hence, with linear algebra in statistics, you are dealing with vectors and matrices, rather than numbers. Since we are talking about vectors and matrices and overlooking scalars or numbers, it is better to understand the terms properly.

  1. Vectors - it is an assembly of numbers and can be found arranged in a row or a column. A vector has just a single index, which can point to a specific value within the Vector                                                                                 

  2. Matrix - is a 2D arrangement of numbers and has two indices. The first points to the row and the second points to a column. A Matrix can have multiple numbers of rows and columns.

  3. Scalar - it is just any number e.g. 45 or 90. 

It is essential to note that a Vector is also a matrix, but with a single row and single column. (Fig .1)

Linear Algebra Equations

As you know by now that linear algebra is a study of linear combinations. You can understand it by the illustrations in the Fig.1 above.  You have become familiar with terms like vector spaces, lines, and planes. All these are in use for linear transformations. Linear transformations also use vectors, matrices, and linear functions.  All of these is an essential part of studies related to linear sets of equations and transformation values.

The General Linear Equation Formula is –

m1x1+m2x2……+ mnxn=k, where, m is the coefficients, x is unknown, and k is the constant.  

Linear Algebra Matrix                         

There also exists a system of linear algebraic expressions known as the linear algebra matrix. By a system of algebraic expression, we mean there is a ‘system’. One can solve this system of linear equations by using the matrices. Matrices represent systems of linear equations. Here it is essential to know that there are three kinds of elementary row operations on matrices:

  1. Adding a multiple of one row to another row

  2. Multiplying all entries of one row by a non-zero constant

  3. Interchanging two rows.

Different Types of Matrices 

As you know, a matrix is made of rows and columns. These rows and columns define the size or dimension of a linear algebra matrix. There are different types of arrays. They get a classification of the following models- row matrix, column matrix, null matrix, diagonal matrix, square matrix, upper triangular matrix, lower triangular matrix, symmetric, and asymmetric matrix.

The Study of Linear Algebra  

Given the foundational relationship of linear algebra with the field of applied machine learning, the impact of linear algebra is important to consider. Some clear marks of linear algebra on statistics and statistical methods include:

  1. Use of vector and matrix notation, especially with multivariate statistics.

  2. Answers to least squares and weighted least squares, such as in linear regression.

  3. Estimates of mean and variance of data matrices.

  4. The covariance matrix plays a key role in multinomial Gaussian distributions.

  5. Principal component analysis for data reduction draws many of these elements together.

Real-Life Applications of Linear Algebra 

The real-life applications of linear algebra have broad applications over a wide range of fields.

  1. Engineering, such as a line of springs.               

  2. Graphs and Networks, such as analysing networks.

  3. Linear Programming, the simplex optimisation method.

  4. Linear Algebra for functions, used widely in signal processing.

  5. Computer Graphics, such as the various translation, rescaling and rotation of images.

  6. Use of linear algebra in automated facial recognition technology.

  7. It is also used in coding theory. One key error-correcting code is known as the hamming code.

FAQ (Frequently Asked Questions)

Q1. Is Linear Algebra Useful?               

The topic of linear algebra is central to almost all areas of mathematics. Linear algebra is in use in most of the science and engineering areas. It allows the modelling of many natural phenomena and inefficient computing based on such models.  Did you know that linear algebra is also the building block for search engine algorithms? Yes, it is this branch of mathematics that helps Google engineers to come up with the method to rank webpages to show them in Google search results pages. The next time you get an accurate and relevant result to your search query, you should thank linear algebra!   

Q2. What Exactly is Linear Algebra?

Linear algebra is an essential part of the Mathematics subject. It is all about linear combinations. One can say that, is about using arithmetic on columns of numbers that are called as vectors and an arrangement of numbers called as matrices. This system helps to create new columns and new assembly of numbers. Linear algebra is the study of vector spaces and mappings that are essential for linear transformations.