Linear Algebra: A Foundational Course for Diverse Disciplines

Linear algebra stands as a cornerstone of modern mathematics, providing essential tools and concepts applicable across a wide spectrum of fields. This article will explore the nature of a typical introductory linear algebra course, its core components, and its relevance to various academic and professional pursuits.

Introduction to Linear Algebra

Linear algebra is an area of mathematics devoted to the study of structure-preserving operators on special sets (linear operators on vector spaces). It is one of the most important and basic areas of mathematics, with many real-life applications. The concepts of linear algebra are extremely useful in physics, economics and social sciences, natural sciences, and engineering. The material we take up in this course has applications in physics, chemistry, biology, environmental science, astronomy, economics, statistics, and just about everything else.

Core Topics in a Linear Algebra Course

A first course in Linear Algebra typically covers the following core topics:

Algebra and geometry of vectors and matrices: This includes operations on vectors and matrices, such as addition, subtraction, scalar multiplication, and matrix multiplication.
Systems of linear equations and Gaussian elimination: This involves solving systems of linear equations using techniques like Gaussian elimination and understanding the different types of solutions (unique, infinite, or no solution).
Eigenvalues and eigenvectors: These concepts are crucial for understanding the behavior of linear transformations and matrices, with applications in areas like stability analysis and data compression.
Gram-Schmidt and least squares: The Gram-Schmidt process provides a method for orthogonalizing a set of vectors, while least squares methods are used to find the best approximate solution to an overdetermined system of linear equations.
Symmetric matrices and quadratic forms: Symmetric matrices have special properties and are important in optimization and other applications. Quadratic forms are functions that can be expressed in terms of a symmetric matrix.
Singular value decomposition and other factorizations: Singular value decomposition (SVD) is a powerful technique for decomposing a matrix into a product of simpler matrices, with applications in dimensionality reduction and image processing. Other factorizations, such as LU and QR decompositions, are also important.

Additional Topics

Depending on the time available and the instructor's preferences, additional topics may be covered, such as:

Markov chains and Perron-Frobenius: Markov chains are mathematical models for systems that evolve over time, while the Perron-Frobenius theorem provides information about the eigenvalues and eigenvectors of certain types of matrices.
Dimensionality reduction: This involves reducing the number of variables in a dataset while preserving its essential structure, with applications in machine learning and data visualization.
Linear programming: This is a technique for optimizing a linear objective function subject to linear constraints, with applications in resource allocation and scheduling.

Course Structure and Delivery

Linear Algebra is not coordinated in the same sense as other multi-section courses with a common final exam (e.g., calculus). As such, the instructor has final discretion in topics chosen and course policies.

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Online Asynchronous Format

Many linear algebra courses are offered online in an asynchronous format. This means that the course is entirely web-based and to be completed asynchronously between the published course start and end dates.

Software Tools

Students can use Free Online OCTAVE version for the purpose of this course or download the MATLAB and Simulink Student Suite software from MathWorks.

Prerequisites

To succeed in this course you will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems. Prerequisites are Math 52 (previously known as Math 1B or N1B), 10B, or N10B. Credit Restriction Courses. Students will receive no credit for this course after completing the course(s) below. -Credit Restrictions Students will receive no credit for MATH 56 after completing MATH 54, MATH N54, or MATH W54. 1 ACT and SAT scores must be submitted to NYU and posted in your academic records in order to satisfy the prerequisite.

Instructor Qualifications

Instructors often possess a Master of Science degree or higher in mathematics or a related field. For example, Dragana is a passionate and experienced mathematics educator with a Master of Science in Industrial Engineering. She brings extensive expertise in both virtual and in-person instruction, creating engaging and effective learning environments for diverse student populations. Dragana is deeply committed to fostering academic excellence, and personal growth in her students. Outside the classroom, she enjoys reading, traveling, hiking, gardening, and playing volleyball.

Relevance to Other Disciplines and Career Paths

Graduates from the Department of Mathematics might take a job that uses their math major in an area like statistics, biomathematics, operations research, actuarial science, mathematical modeling, cryptography, or mathematics education. Or they might continue into graduate school leading to a research career. Professional schools in business, law, and medicine appreciate mathematics majors because of the analytical and problem solving skills developed in the math courses.

Linear algebra's principles underpin various fields, including:

Computer Science: Essential for graphics, image processing, machine learning, and algorithm design. MAS 4115 Linear Algebra for Data Science 3 Credits A second course in linear algebra, focusing on topics that are the most essential for data science. Introduces theory and numerical methods required for large data-sets and machine learning. Topics include LU, QR, and singular-value decompositions; conditioning and stability; the DFT and filters; deep learning; fully connected and convolutional nets.
Physics and Engineering: Used extensively in mechanics, electromagnetism, quantum mechanics, and signal processing.
Economics: Applied in econometrics, optimization, and game theory.
Statistics: Forms the basis for regression analysis, multivariate analysis, and data mining.

Related Mathematics Courses

The study of linear algebra often complements and builds upon concepts introduced in other mathematics courses. Here's a glimpse of courses that relate to and expand upon linear algebra principles:

Calculus: MAC 2311 Analytic Geometry and Calculus 1 4 Credits In this course, students will develop problem solving skills, critical thinking, computational proficiency, and contextual fluency through the study of limits, derivatives, and definite and indefinite integrals of functions of one variable, including algebraic, exponential, logarithmic, and trigonometric functions, and applications. Topics will include limits, continuity, differentiation and rates of change, optimization, curve sketching, and introduction to integration and area. MAC 2312 Analytic Geometry and Calculus 2 4 Credits Techniques of integration; applications of integration; differentiation and integration of inverse trigonometric, exponential and logarithmic functions; sequences and series. (Note: Credit will be given for at most one of MAC 2312, MAC 2512 and MAC 3473.) MAC 2313 Analytic Geometry and Calculus 3 4 Credits Solid analytic geometry, vectors, partial derivatives, multiple integrals. (Note: Credit will be given for at most one of MAC 2313 and MAC 3474.)
Differential Equations: MAP 2302 Elementary Differential Equations 3 Credits First-order ordinary differential equations, theory of linear ordinary differential equations, solution of linear ordinary differential equations with constant coefficients, the Laplace transform and its application to solving linear ordinary differential equations. (M)
Discrete Mathematics: MAD 3107 Discrete Mathematics 3 Credits Logic, sets, functions; algorithms and complexity; integers and algorithms; mathematical reasoning and induction; counting principles; permutations and combinations; discrete probability. Advanced counting techniques and inclusion-exclusion.
Abstract Algebra: MAS 4301 Abstract Algebra 1 3 Credits Sets and mappings, groups and subgroups, homomorphisms and isomorphisms, permutations, rings and domains, arithmetic properties of domains, and fields. Requires facility in writing proofs. MAS 4302 Abstract Algebra 2 3 Credits A second course in Abstract Algebra, focusing on Galois Theory, the algebraic theory of fields and polynomial equations. Introduces concepts of abstract algebra used in settling famous historical problems including the problems of angle trisection and duplication of cubes by ruler and compass constructions, and the insolubility of polynomial equations of the fifth degree by radicals.
Real Analysis: MAA 4102 Introduction to Real Analysis 1 3 Credits Theory of real numbers, functions of one variable, sequences, limits, continuity and differentiation; continuity and differentiability of functions of several variables. Those who plan to do graduate work in mathematics should take MAA 4211. Credit will be given for, at most, one of MAA 4102, MAA 4211, or MAA 5104. MAA 4103 Introduction to Real Analysis 2 3 Credits Continues the advanced calculus for engineers and physical scientists sequence. Theory of integration, transcendental functions and infinite series. MAA 4102 is not recommended for those who plan to do graduate work in mathematics; these students should take MAA 4212. Credit will be given for, at most, one of MAA 4103, MAA 4212 and MAA 5105. MAA 4211 Real Analysis and Advanced Calculus 1 3 Credits Advanced treatment of limits, differentiation, integration and series. Includes calculus of functions of several variables. Credit will be given for, at most, one of MAA 4211, MAA 4102 and MAA 5104. MAA 4212 Real Analysis and Advanced Calculus 2 3 Credits Continues the advanced calculus sequence in limits, differentiation, integration and series. Credit will be given for, at most, one of MAA 4212, MAA 4103 and MAA 5105. MAA 4226 Introduction to Modern Analysis 1 3 Credits Topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables, Stokes' theorem and the Lebesgue theory. Credit will be given for, at most, MAA 4226 or MAA 5228. MAA 4227 Introduction to Modern Analysis 2 3 Credits Continues the modern analysis sequence discussing the topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables, Stokes' theorem and the Lebesgue theory. Credit will be given for, at most, MAA 4227 or MAA 5229.

Textbooks

This course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Linear Algebra. Each unit has been further divided into a sequence of sessions that cover an amount you might expect to complete in one sitting. Each session has a video lecture on the topic, accompanied by a lecture summary.

Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley, MA: Wellesley-Cambridge Press, February 2009.
Strang, Gilbert. Introduction to Linear Algebra. 5th ed. Wellesley, MA: Wellesley-Cambridge Press, February 2016.

Time Commitment

MIT expects its students to spend about 150 hours on this course. More than half of that time is spent preparing for class and doing assignments.

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