Linear algebra is one of the fundamental branches of mathematics that deals with studying linear equations and their representation in the vector space using matrices. It is the study of vector spaces and linear transformations.
Linear functions, the system of linear equations, vector spaces, vectors, matrices, and linear transformation are the critical concepts in this branch of mathematics. Broadly, vectors are elements that we can add, and linear functions are functions of vectors that respect vector addition. A matrix emerges when the information related to linear functions is arranged in an organized form.
Linear algebra is one of the most important mathematical tools. It is essential in pure and applied mathematics alike. Linear algebra is also widely applicable in several fields such as physics, engineering, economics, computational science, and natural sciences. Linear algebra is one of the most valuable branches of mathematics today. An introduction to linear algebra is the building block to understanding many unknowns in the sciences.
In vector geometry, two or more vectors are orthogonal if they are perpendicular to each other. In other words, the dot product they yield is always zero.
A vector space V is a collection of objects with a (vector) defined as having two operations, addition and scalar multiplication. Vector addition combines two vectors, u and v, into a single vector u + v. Scalar multiplication is a way of combining scalar, k, with a vector, v, to end up with the vector kv.
Vector spaces are subject to the following ten axioms:
A vector subspace S of a vector space V is a nonvoid subset of V and is a vector space in its own right, following under defined operations of the same addition and scalar multiplication. A subspace S of a vector set V is a working set that allows the original data set V to shrink into a smaller data set S.
Row vector and column vector are part of a rectangular array of values or elements. A matrix M has a rows and b columns. A vector v, when treated as a matrix, has either one row or one column.
Let us understand what is a row vector and a column vector.
If every element of a vector space can be written as a linear combination of some vectors and the vectors are independent of each other it is a basis. A basis must fulfil two conditions. First, it should have linear independence. Secondly, they must span the whole space.
Now, let us understand what is an orthonormal basis. A basis is orthonormal if all of its vectors have a norm (length) of 1. Additionally, all the vectors should be orthogonal (perpendicular) to each other with an inner product of 0.
Let us begin by learning what exactly a matrix is. In linear algebra, a matrix is a rectangular array of numbers, symbols or expressions (or other mathematical elements). It means that a matrix is an 𝑚 × 𝑛 array of scalars from a given field 𝐹. These are called elements or entries and are arranged in rows or columns. We can identify each entry by the row and column in which it lies. When the number of rows is equal to the number of columns of a matrix, it is a square matrix.
Let’s move on to exploring matrix operations. There are two primary matrix operations: matrix addition and scalar multiplication of matrices. In matrix addition, if matrices A and B are of the same size, the sum of the matrices is a matrix. It is important to note that only matrices of the same size can be added. Additionally, the notion of subtracting matrices is similar to matrix addition. In this case, matrix subtraction is the operation of subtracting two matrices of the same size.
In matrix scalar multiplication, if a matrix 𝐴 is multiplied by a scalar 𝑘, then all the matrix elements will be multiplied with the scalar. Matrix multiplication is a vital operation in linear algebra. It is the process of multiplying matrices. Matrix multiplication is possible only if two matrices (𝐴 and 𝐵) are compatible, i.e., the number of columns in 𝐴 is equal to the number of rows in 𝐵. Further, matrix multiplication is not commutative, meaning that the order of matrices is critical in this operation.
Quadratic forms like linear functions have matrix representation. When matrix 𝐴 denotes an 𝑛 x 𝑛 symmetric matrix with entries and x represents an 𝑛 x 1 column vector, Q = x’𝐴x is said to be a quadratic form.
Quadratic forms can be classified as follows:
Linear algebra is one of the most vital branches of pure and applied mathematics. It deals with vector spaces and linear transformation. The central objective of linear algebra is to analyze the linear functions in a finite-dimensional vector space.
A vector is an object that has a direction and a magnitude. A vector space is a collection of vectors following vector addition and scalar multiplication and satisfies the requirements of vector axioms.
A linear transformation or linear map is a function 𝐓: 𝐑ⁿ→𝐑m that satisfies the operations of vector addition and scalar multiplication. In this case, 𝐑ⁿ is the domain, 𝐑m is the codomain, and 𝐓 is the mapping between two vector spaces. Linear transformations preserve the structure of vector spaces.
Matrices are used in almost every branch of mathematics and mainly in linear algebra, statistics, and geometry.
Matrices (plural form of a matrix) are integral to mathematics, particularly linear algebra, statistics, and geometry. A matrix is an 𝑚×𝑛 array of scalars from a given field 𝐹. Here, 𝑚 represents the number of rows and 𝑛 represents the number of columns. The number of rows and columns a matric has is its matrix dimension or matrix order.
Matrices are helpful in velocity, acceleration, and linear dynamics equations.
In linear algebra, an inverse matrix is when the matrix 𝐴 is invertible, and there exists an inverse of 𝐴 that is 𝐴⁻¹. An inverse matrix is represented as 𝐴⁻¹𝐴 = 𝐼, and 𝐴𝐴⁻¹ = 𝐼.
Linear algebra deals with several basic concepts and operations. Here we explore some of them.
When 𝑉 is a finite-dimensional vector space, the dimension of 𝑉 (dim 𝑉) is the size of any basis of 𝑉.
In linear transformations, eigenvectors are non-zero vectors whose direction remains unchanged by the linear transformation. Eigenvectors are also known as characteristic vectors. The eigenvalue is the scalar λ of a matrix and is alternatively known as the characteristic root/value. An eigenvector scales according to the factor of the eigenvalue. The function is 𝐴x = λx, where 𝐴 is the matrix, x is the eigenvector corresponding to the eigenvalue λ.
Linear Algebra courses are essential because of their applicability across disciplines and fields. Linear algebra and its applications have led to immense growth in demand for linear algebra courses. The applications of linear algebra in real life are as diverse as their applicability in theory. Introduction to linear algebra can open many doors across several fields for students and professionals alike.
Linear algebra courses are available in both online and offline learning modes. Linear algebra classes can take place either online or offline. Let us look at what the two methods of learning entail.
Linear Algebra courses online are the way of the future. They offer several advantages over offline courses. Let’s explore some of the benefits of an online Linear Algebra course compared to an offline course.
A Linear Algebra course covers several important aspects which have application across fields. Some of the best Linear Algebra courses provide a great introduction to Linear Algebra. It begins with a comprehensive syllabus in linear algebra training, especially for beginners.
Here we outline the concepts and topics of a Linear Algebra course.
There are Linear Algebra courses that are available for specific sectors or sections of people as well. Here is a list.
Linear Algebra is one of the most relevant branches of mathematics today. Linear Algebra and its applications have become essential for professional growth across many sectors. Even beginners and mathematics enthusiasts are turning to the fundamentals of linear algebra through available linear algebra courses. Free linear algebra courses and linear algebra courses online are becoming hugely popular.
The applications of linear algebra are widespread in mathematics, physics, economics, computer science, engineering, artificial intelligence, machine learning, and data science, to name only a few. A linear algebra certification can boost professional careers due to the versatility of this branch of mathematics. Knowledge of linear algebra is an asset because of its transferability across diverse professions. The potential of this industry is immense as the sectors rely on linear algebra for continuous growth.
Linear Algebra courses in India have seen a growth in demand. Linear algebra as a branch in mathematics has become immensely popular and a valuable asset for professionals in the fields of science and engineering. Linear algebra and its diverse applications make it a versatile knowledge transferable to several industries. The fundamentals of linear algebra can help in many industries, ranging from the classical fields of physics, economics, and engineering to emerging industries like artificial intelligence, machine learning, and quantum mechanics.
Several offline and online linear algebra courses are flourishing in India today, including free linear algebra courses and linear algebra certification courses. The accelerating demand for linear algebra courses in India indicates its value for educational and career growth. It is a high-demand course that is part of the curriculum of some of the top universities in the world. The fundamentals of linear algebra have become a part of the syllabus within many of the most popular courses and fields available today.
Specific courses on linear algebra deal with specialized subjects as well, such as applications of linear algebra in engineering, linear algebra geometric approach, computational linear algebra, linear algebra and numerical analysis, linear algebra with python, linear algebra and complex analysis, engineering statistics and linear algebra, linear algebra and optimization for machine learning, elementary linear algebra a matrix approach. These are just a few specialized linear algebra courses that are available. The demand for these courses are accelerating, too. The future of linear algebra looks very bright.
Linear Algebra as a skill is a valuable asset among freshers and professionals in the current job market. A Linear Algebra Specialist can find employment across several industries and fields, such as data analytics, engineering, artificial intelligence (AI), quantum mechanics, physics, economics, and mathematics. The knowledge and understanding of this specific branch of mathematics are foundational for many professions and hence, transferable across them.
The pay package of a Linear Algebra Specialist can vary depending on several factors. Here we outline a few factors that affect the salary of a Linear Algebra Specialist in India.
The salary of a Linear Algebra Specialist can vary from country to country. Let us look at the factors that can affect their salary package.
Average Salary Hike
Analyse movie data from the past 100 years and find out various insights to determine what makes a movie do well.
Solve a real industry problem through the concepts learnt in exploratory data analysis
Build a model to understand the factors on which the demand for bike sharing systems vary on and help a company optimise its revenue
Help the sales team of your company identify which leads are worth pursuing through this classification case study
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Build a machine learning model to identify fraudulent credit card transactions
Forecasting the sales on the time series data of a global store
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This is an ETL project which will cover the topics like Apache Sqoop, Apache Spark and Apache Redshift
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With the IPL season commencing, let's go ahead and do an exciting assignment on sports analytics in Tableau.
Build a regularized regression model to understand the most important variables to predict the house prices in Australia.
Analyse the dataset of parking tickets
Practice MapReduce Programming on a Big Dataset.
In this module, you will solve an industry case study using optimisation techniques
This module will contain practice assignment & all resources related to a classification based problem statement.
Linear Algebra is a fundamental branch of mathematics that deals with linear equations and their representation in vector space through matrices. It deals with linear equations, matrices, determinants, vector spaces, and linear transformations.
A vector space 𝑉 is a collection of objects called vectors and is defined by the twin operations of vector addition and scalar multiplication. It is subject to ten vector space axioms. A vector subspace is a subset of vector space when the subspace is closed under operations of vector addition and scalar multiplication and satisfies all ten vector space axioms.
A matrix is an array of numbers, expressions, or symbols comprising of 𝑚 number of rows and 𝑛 number of columns. An uppercase letter represents a matrix, and the mathematical objects of the rows and columns are called entries. A matrix is subject to two standard operations: matrix addition and multiplication with a scalar. The number of rows and columns of a matrix represents the matrix size.
Linear algebra is important in various areas of science. Since linear equations are easy to solve, practically all areas of modern science comprise models wherein equations are approximated using linear equations. The solution of these equations facilitates the development of the theory. Moreover, linear algebra lets you categorize and understand objects, spaces, and situations. A noteworthy theorem in Hilbert Spaces helps you to recognize any distinguishable Hilbert space to a certain specific Hilbert Space.
In unsupervised learning, models are trained through the unlabeled dataset, and they act on the data without any supervision. Unsupervised Learning algorithms for which linear algebra can be used are Single Value Decomposition (SVD), Clustering, and Components Analysis. In these algorithms, linear algebra helps to determine the insights and patterns from the specified data.
It is recommended to adopt the result-first approach if you aim to learn linear algebra for machine learning. In this approach, you begin by learning to practice a predictive style end-to-end problem through a tool known as sci-kit-learn or Pandas in Python. The particular process will offer a skeleton that helps you know how to gradually extend your knowledge on the working of an algorithm and math that supports all of these algorithms.
Yes, linear algebra contains arithmetic operations and is therefore used in arithmetic. The knowledge of how to add, subtract and multiply vectors, scalars, and matrices is important. Most of the functions are implemented and provided through the API Call in Modern Linear Algebra Libraries. Moreover, an understanding of matrix operations and vector and Matrix notation is required to accomplish different arithmetic operations.
To learn statistics, you should learn linear algebra, specifically multivariate statistics. Note that mathematics is one of the key pillars that support Machine learning and statistics. To read and decode data, you have to learn the operations and notations of Linear algebra. Contemporary statistics use both tools and notation of linear algebra to illustrate the techniques and tools of statistical methods. Many of the complex formulas of linear algebra will help you solve problems in statistics.
Linear algebra demands less brain work than Calculus and is simpler than elementary calculus. Usually, it is the foremost actual math course that students take up. Since it would be a significant change of pace from the standard high school mathematics to calculus, it can seem difficult to students initially. In cases when you need to know definitions and terms for determining which calculations are required to get the desired result, linear algebra may seem a bit difficult.
Linear algebra is used in Machine Learning, Least Square approximation, Traffic Flow, Electrical Circuits, Determinants, Genetics, Graph Theory, Cryptography, Markov Chain, Pattern Recognition, Data Classification, and Clustering, Singular Value Decomposition for recommendation systems, etc. It is also used in Graphics Programming, Image processing, Prediction Algorithms, Computer Vision, Quantum Computation, Web Search, Information Retrieval, etc.
Machine learning is the programming to optimize for providing the best possible outcome. Math is required to know how the problem is solved. The first step in the process of learning Math for machine learning is to learn linear algebra. The knowledge of linear algebra helps you to solve the problem of data and computations in machine learning models. It involves matrices, vectors, and tensors which are quite useful for solving machine learning problems.
Linear Algebra is essential for statistical programming. Although computers automatically do calculations, they can’t interpret the type outcomes of our calculations i.e. mathematical or statistical. For that, you must know linear algebra. Secondly, almost all the analysis in mathematics needs some sort of linear algebra. Thirdly, games are developed using the principle of linear and matrix transformations and knowledge of linear algebra. You can solve advanced mathematical and statistical calculations in programming using linear algebra.
One of the widespread real-life applications of linear algebra is it helps create ranking algorithms in search engines like Google. Other examples include analysis of digital signals for audio or video applications, correction of codes that have been tampered with during transmission or processing, checking the energy levels of atoms, checking the distribution of microwave energy in a microwave oven, space study, etc. Linear algebra is also used for correcting the code that was used for projecting a 3D view into a 2D plan, controlled by linear maps.
The common and important terms used in advanced linear algebra are Linear Transformations (transforms a function from one vector space to another), Inverse of a Matrix (an inverse of a matrix is multiplied with the original matrix), Eigenvector (a non-zero vector that changes by a scalar factor), Linear Map (mapping that preserves vector multiplication and vector addition).
Applied linear algebra is generally introduced to students at a graduate level in the fields of engineering, physics, and applied mathematics. It is the branch of algebra that integrates the concepts of elementary as well as advanced linear algebra with their real-world implications. Concepts like the norm of a vector, Schur's complement of a matrix, QR factorization, etc. are covered in Applied Linear Algebra.
Linear algebra deals with linear equations and their representations in the vector space with the help of matrices. It is concerned with vectors, vector spaces, linear functions, matrices, and the system of linear equations. All these concepts are a prerequisite for learning sister topics like functional analysis and geometry.
Linear algebra is a vital branch of mathematics that has many applications. Linear algebra is widely applicable across fields such as mathematics, physics, economics, engineering, physics, data sciences, computational analysis, artificial intelligence, and machine learning, to name a few. As a result, knowledge of linear algebra is a valuable asset for students and professionals.