Linear Algebra Course Overview

    What is Linear Algebra?

    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. 

    What does it mean for vectors to be orthogonal?


    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. 

    What is a vector subspace?


    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: 

    1.  Closed under addition: u + v is in V
    2.  Addition is commutative: u + v = v + u.
    3.  Addition is associative: (u + v) + w = u + (v + w) 
    4.  Additive identity 0 (called the zero vector) e
    5. xists: u + 0 = u. 
    6.  Additive inverse −v exists: u + (−u) = 0. 
    7.  Closed under scalar multiplication: cu is in V
    8.  Distributive law for vectors: c(u + v) =cu+cv. 
    9.  Distributive law for scalars: (c + d)u = cu + du. 
    10.  Multiplication is associative: (cd)u = c(du). 
    11.  Multiplicative identity exists: 1u = u.

    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

    A row vector vs. A column vector

    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. 

    1. A row vector is an ordering of collection of numbers written in a row or horizontally. Example of a row vector is: x=x₁, x₂,........xₙ.
    2. A column vector is an ordering of collection of numbers written in a column (vertically). 

    What is an orthonormal basis?

    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.

    Matrix Operations

    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.

    Eigenvalues and Eigenvectors

    Eigenvectors are non-zero vectors that do not change their orientation but only their scale depending on their corresponding eigenvalue. A scalar λ is an eigenvalue of matrix 𝐴 if there is a solution x of 𝐴x = λx; such an x is called an eigenvector corresponding to λ.

    Similarity Transformations

    The similarity transformation converts the matrix representation of a general linear transformation from one frame to another. The following equation can define similarity transformation: 𝐴 = 𝑃⁻¹𝐵𝑃. Here, 𝐴, 𝑃, and 𝐵 are square matrices. 𝐴 and 𝐵 are similar if we find a non-singular 𝑛×𝑛 matrix 𝑃 which fulfills the 𝐴 = 𝑃⁻¹𝐵𝑃 equation.

    Quadratic Forms

    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:

    1. negative definite: Q < 0 when x ≠ 0.
    2. negative semidefinite: Q ≤ 0 for all x and Q = 0 for some x ≠ 0.
    3. positive definite: Q > 0 when x ≠ 0.
    4. positive semidefinite: Q ≥ 0 for all x and Q = 0 for some x ≠ 0.
    5. indefinite: Q > 0 for some x and Q < 0 for some other x.

    Linear Algebra

    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

    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.

    Inverse Matrix


    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.


    The basics of linear algebra

    Linear algebra deals with several basic concepts and operations. Here we explore some of them. 

    • Vectors are one of the primary components of linear algebra. A vector is a mathematical object which has magnitude and direction. It follows the operations of vector addition and scalar multiplication. There are different types of vectors. However, you cannot add the different types of vectors. 
    • Matrices are the plural form of a matrix. A matrix includes symbols, numbers, or expressions arranged across 𝑚 number of rows and 𝑛 number of columns. It has two indices: the first index points to the row and the second to the column. 
    • A linear transformation is generally given by matrix multiplication. A linear transformation is a function 𝐓: 𝐑ⁿ→𝐑m which satisfies the conditions: 
    • T(x+y)=T(x)+T(y)
    • T(𝑎x)=𝑎T(x)
    • Linear independence is a central concept in linear algebra. When two or more vectors cannot be written as a combination of the other vectors, they are linearly independent. It is a property of a set of vectors. 
    • When 𝑆 is a subset of a vector space 𝑉, span (𝑺) is the smallest subspace of 𝑉 that contains 𝑆. Subspace is a subset of a vector space 𝑉 when it is non-empty and fulfills the linear operations of the vector space.
    • A linearly independent spanning set for vector space 𝑉 is a basis. A basis is a subset (𝑆) of vector space (𝑉) if it follows that:
    • 𝑆 spans 𝑉.
    • 𝑆 is linearly independent in 𝑉.

    When 𝑉 is a finite-dimensional vector space, the dimension of 𝑉 (dim 𝑉) is the size of any basis of 𝑉.

    • Application of Dimension to Binary Space

    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.

    Why is an online Linear Algebra Course better than Offline Linear Algebra Course?

    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.

    • Offline courses in linear algebra are held in physical classrooms and generally follow a fixed schedule. The teacher largely determines the pace of learning. Communication between students and teachers takes place in person. 
    • Online courses in linear algebra use digital tools and methods of teaching. The classes are virtual and independent of fixed space and time. The students primarily determine the pace of learning as the resources are virtually available. 

     

    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. 

    • Flexibility- Online courses offer more flexibility to students. Online classes are accessible from anywhere and at any time. There are no fixed schedules that the students need to follow or a physical location that they need to be. 
      1. Cost-effective- Online courses are more affordable than offline courses. 
      2. Time consideration- Online linear algebra courses do not require a student to be at a fixed location, saving commuting time. 
    • Increases choice- The choice of courses available through online learning mode is incomparable to the offline ones. Prospective students can opt from the gamut of options available from any university or college worldwide without being held back by its location or timeline. 
    • Materials are reusable- Course material is readily accessible to the students who have the option of reusing it multiple times. The use of digital tools allows students to download resources and lectures for future reference. 
    • The pace of learning- In an online Linear Algebra course, the students determine the learning speed rather than the course instructor. It is helpful for beginners, working professionals, and mathematics enthusiasts to learn linear algebra

    Linear Algebra Course Syllabus

    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.

    • Systems of linear equations
    • Matrices
    • Matrix operations: Addition, Scalar Multiplication, Matrix Multiplication, Identity, Transposes
    • Row operations, including Inverse of a Matrix 
    • Elementary Matrices
    • Equivalent Matrices
    • Determinants: Definition, Properties, Cofactor Expansion, and Cramer’s Rule
    • Vector spaces: Span and Subspaces 
    • Linear Combination
    • Linear Dependence and Independence
    • Null space and Column space
    • Basis and Dimension
    • Homogeneous Systems
    • Coordinate Systems and Isomorphisms
    • Rank of Matrix
    • Change of Basis
    • Inner Product Spaces: Inner Product, Length, and Orthogonal Vectors
    • Cauchy-Schwarz Inequality
    • Orthonormal (Orthogonal) Basis
    • Gram-Schmidt Process
    • Orthogonal Projections
    • Linear Transformations: Definitions and Properties
    • Kernel and Range of a Linear Transformation
    • Matrix of a Linear Transformation
    • Eigenvalues and Eigenvectors
    • Diagonalization in Linear Algebra
    • Symmetric Matrices and Orthogonal Diagonalization
    • Linear Models and Least-Squares problems

    There are Linear Algebra courses that are available for specific sectors or sections of people as well. Here is a list.

    • Linear Algebra for Machine Learning
    • Linear Algebra for Data Science
    • Linear Algebra for Computer Science
    • Linear Algebra for Engineers
    • Linear Algebra in Economics
    • Linear Algebra for Quantum Mechanics
    • Linear Algebra in Artificial Intelligence
    • Linear Algebra in Physics
    • Linear Algebra for Beginners

    Projecting Linear Algebra Industry Growth in 2022-23

    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.

    The Accelerating Demand for Linear Algebra Courses in India

    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 Specialist Salary in India

    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. 


    High demand for linear algebra as a skill ensures that Linear Algebra Specialists command a lucrative pay package. The average annual salary of a Linear Algebra Specialist in India is INR 13,50,000 per annum. Their salary ranges between INR 10,00,000 and INR 20,70,000 per annum.

    Factors on which Linear Algebra Specialist salary in India depends

    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.

    • Place of Employment: The city or country of employment plays a critical role in the pay package of a Linear Algebra Specialist. The location of the job determines the cost of living, cost of real estate, and quality of life parameters of a specialist. As a result, the salary package reflects these factors. 
    • Experience: The salary of a Linear Algebra Specialist is a reflection of their experience in the industry. Higher industry experience is generally proportional to a higher salary package.
    • Sector or Field of Employment: A Linear Algebra Specialist is employable across several sectors and fields. The skills and knowledge of linear algebra are transferable as it is a fundamental branch of mathematics. A Linear Algebra Specialist may find employment as a data analyst or scientist, machine learning expert, engineer, economist, and AI expert, to name a few. Therefore, the sector or the field of employment can affect the salary package of a Linear Algebra Specialist.
    • Skill Level or Proficiency: The level of expertise of a Linear Algebra Specialist plays a crucial role in determining the remuneration package. Higher proficiency or skill level commands a higher salary package in the job market.

    Linear Algebra Specialist Starting Salary in India

    The basic salary for Linear Algebra specialist in India is estimated to start from INR 10 Lakhs annually. The salary grows with experience, skills and other contributing factors influencing work efficiency of the candidate.

    Linear Algebra Specialist Salary Abroad

    The average annual compensation offered to Linear Algebra analysts abroad is estimated to lie near USD 150k to USD 201k per annum. These numbers depend upon the position and experience a candidate holds, which is prone to go higher with time. Leading brand names such as Apple, Intel, Amazon, and Microsoft offer lucrative opportunities to Linear Algebra specialists.

    Linear Algebra Specialist Starting Salary Abroad

    A Linear Algebra specialist based in abroad is estimated to have a starting salary of around USD 40k to USD 90k, which is likely to grow with candidate’s experience in the field. Compared to salary offered to Linear Algebra Specialist in India, abroad offers a significantly higher sum.

    Factors on which Linear Algebra Specialist Specialist Abroad salary depends

    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.

    • Country of Employment: The cost of living varies from one country to another. The salary package reflects the variance in the living costs of a Linear Algebra Specialist depending on their location of employment.
    • Work Experience: As in most professions, work experience is one of the crucial factors in determining salary. 
    • Field of Employment: Linear Algebra Specialists are employable across many professions. Some industries pay more than others. So, where you find employment plays a significant role in determining the salary package.
    • Skills: Proficiency level in linear algebra is vital to determining the salary package of a Linear Algebra Specialist. Higher skill levels generally translate to higher remunerations. 
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    Frequently Asked Questions about Linear Algebra Course

    What is Linear Algebra?

    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. 

    What is a vector space?

    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.

    What is a matrix?

    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. 

    Why study Linear Algebra?

    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.

    Can linear algebra be used for unsupervised learning?

    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.

    How to get started with linear algebra for machine learning?

    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.

    Is linear algebra used in arithmetic?

    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.

    Is the knowledge of linear algebra essential for statistics?

    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.

    Is Linear Algebra difficult?

    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.

    What are the applications of linear algebra?

    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.

    Does machine learning use linear algebra to solve data problems?

    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.

    Why Linear Algebra Is Important For Programming?

    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.

    What are the real-life applications of 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.

    What are the important terms used in advanced linear algebra?

    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).

    What is Applied Linear Algebra?

    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.

    What is linear algebra concerned with?

    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.

    Why is Linear Algebra important?

    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.