Mastering Linear Algebra: An Introduction with Applications

Course No. 1056
Professor Francis Su, PhD
Harvey Mudd College
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Course No. 1056
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What Will You Learn?

  • numbers Get a grasp on the essential concepts and problem-solving methods of first-semester college linear algebra.
  • numbers Understand how to exploit the complimentary geometric and algebraic components of linear algebra.
  • numbers Learn how to solve problems in a host of applications.
  • numbers Study the fine art of proving conjectures in mathematics.

Course Overview

Linear algebra may well be the most accessible of all routes into higher mathematics. It requires little more than a foundation in algebra and geometry, yet it supplies powerful tools for solving problems in subjects as diverse as computer science and chemistry, business and biology, engineering and economics, and physics and statistics, to name just a few. Furthermore, linear algebra is the gateway to almost any advanced mathematics course. Calculus, abstract algebra, real analysis, topology, number theory, and many other fields make extensive use of the central concepts of linear algebra: vector spaces and linear transformations.

Mastering Linear Algebra: An Introduction with Applications is the ideal starting point for this influential branch of mathematics, surveying the traditional topics of a first-semester college course in linear algebra through 24 rigorous half-hour lectures taught by Professor Francis Su of Harvey Mudd College. A multi-award-winning math educator, Professor Su was named “the mathematician who will make you fall in love with numbers” by WIRED magazine.

Linear algebra provides insights into complex phenomena that are part of our daily lives, making them less mysterious and showing the astonishing reach of mathematics in areas such as:

  • Computer Graphics: The field of 3-D computer graphics exists because of linear algebra, which transforms shapes in 3-dimensional space by matrix multiplication.
  • GPS: A Global Positioning Satellite (GPS) receiver, such as a smartphone, determines its position based on time signals from several satellites. Linear algebra shows how to take this seemingly complicated problem and make it accessible.
  • Search Engines: The ability to find information quickly on the internet is a key feature of modern life, and it’s made possible by linear algebra, which keeps track of which nodes on a network are linked, and highlights structures that enable the ranking of important web pages.
  • Recommender Systems: Most of us have experienced websites that seem to know more about our tastes than our own family members. The ability of linear algebra to reveal hidden structures lies behind many of these recommender systems.

Indeed, linear algebra has become so central to our modern data-driven world that more and more educators believe the subject should be introduced earlier in the mathematics curriculum. Linear algebra has spawned truly subtle and sophisticated problem-solving strategies that are favored by specialists, but the underlying concepts are relatively simple and within reach of anyone with a firm grasp of algebra and some analytic geometry. (A background in calculus is helpful, but not required.)

In Mastering Linear Algebra, Professor Su puts a premium on visualizing both the results and the reasoning behind important ideas in linear algebra, giving a geometric picture of how to understand matrices and linear equations. Focusing on a wide range of interesting applications, he works through problems step by step, introducing key ideas along the way, starting with:

  • Vectors and vector spaces,
  • Dot products and cross products,
  • Matrix operations, and
  • Linear transformations and systems of linear equations.

Armed with these essential concepts, you dig deeper into properties and problem-solving strategies involving:

  • Bases and determinants,
  • Eigenvectors and eigenvalues,
  • Orthogonality,
  • Markov chains, and much more.

What Is Linear Algebra?

While the term “linear algebra” may evoke a stark image of straight lines and the manipulation of symbols, the subject is far more elegant than that. The “linear” part refers to linear systems of equations and their geometric manifestations as planes or hyperplanes. In such equations, polynomials with exponents and other nonlinear terms are not present. This makes dealing with equations pleasingly straightforward.

Vectors enter the picture because the linear equations can be viewed as a transformation of one vector into another. And the matrices are arrays of numbers that are the coefficients of these linear equations. The “algebra” part of linear algebra is simply the rules for performing operations on the vectors and matrices. From these basic ideas, a vibrant mathematical universe emerges—a rich interplay between algebra and geometry, between computation and visualization, between the concrete and the abstract, and between utility and beauty.

In the very beginning, Professor Su introduces four themes that you encounter throughout the course:

  • Linearity is a fundamental idea in mathematics and in the world. The idea of linearity arises everywhere—from adapting a recipe to calculating the age of the universe. In linear algebra, this property is embodied by linear transformations, which are functions that change one vector in a vector space into another.
  • To understand nonlinear things, we approximate them by linear things. Many phenomena are nonlinear (think of the motion of planets around the sun), but at small scales they are approximately linear. This idea is the heart of calculus, which uses ideas from linear algebra to approximate nonlinear functions by linear ones.
  • Linear algebra reveals hidden structures that are beautiful and useful. Much of what linear algebra does is uncover hidden structures that give insight into what is really going on in a problem, allowing it to be solved with surprising ease. Seeing these unexpected connections and shortcuts can be an aesthetic experience.
  • Linear algebra’s power often comes from the interplay between geometry and algebra. The effectiveness of linear algebra is due in large part to the way problems can be envisioned in both geometric and algebraic terms. The geometric picture feeds intuition about what a solution might look like, while the algebraic tools show the way to an answer.

Big Data, Tamed

Anyone excited about diving into the vast sea of data made possible by the internet and today’s nearly limitless computing power should definitely study linear algebra. Professor Su covers the math behind several techniques that both tame and exploit big data. Early on, he spotlights the problem of error detection, which is used to identify and correct corrupted computer bits. Then later, he zeroes in on the tricks used to encode data as efficiently as possible—in this case, the JPEG image-compression algorithm. In a look at singular value decomposition, he presents another method of data compression. And, Professor Su considers the challenges of search engines and speech recognition programs, explaining how Markov chains model the probability of what to expect given the current state of a system.

Mastering Linear Algebra also briefly introduces you to quantum mechanics, the notoriously baffling theory of subatomic particles. Since quantum theory is written in the language of vectors and matrices, you need linear algebra to understand it. Professor Su provides a taste of that understanding by showing how the apparently paradoxical superposition of states—in which quantum entities can be in two states at the same time—makes perfect sense when you think of it in terms of linear algebra (specifically, as a linear combination of states in a vector space). You learn this fascinating lesson in Lecture 3—by which point you will already be looking at the world in a whole new way.

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24 lectures
 |  Average 30 minutes each
  • 1
    Linear Algebra: Powerful Transformations
    Discover that linear algebra is a powerful tool that combines the insights of geometry and algebra. Focus on its central idea of linear transformations, which are functions that are algebraically very simple and that change a space geometrically in modest ways, such as taking parallel lines to parallel lines. Survey the diverse linear phenomena that can be analyzed this way. x
  • 2
    Vectors: Describing Space and Motion
    Professor Su poses a handwriting recognition problem as an introduction to vectors, the basic objects of study in linear algebra. Learn how to define a vector, as well as how to add and multiply them, both algebraically and geometrically. Also see vectors as more general objects that apply to a wide range of situations that may not, at first, look like arrows or ordered collections of real numbers. x
  • 3
    Linear Geometry: Dots and Crosses
    Even at this stage of the course, the concepts you've encountered give insight into the strange behavior of matter in the quantum realm. Get a glimpse of this connection by learning two standard operations on vectors: dot products and cross products. The dot product of two vectors is a scalar, with magnitude only. The cross product of two vectors is a vector, with both magnitude and direction. x
  • 4
    Matrix Operations
    Use the problem of creating an error-correcting computer code to explore the versatile language of matrix operations. A matrix is a rectangular array of numbers whose rows and columns can be thought of as vectors. Learn matrix notation and the rules for matrix arithmetic. Then see how these concepts help you determine if a digital signal has been corrupted and, if so, how to fix it. x
  • 5
    Linear Transformations
    Dig deeper into linear transformations to find out how they are closely tied to matrix multiplication. Computer graphics is a perfect example of the use of linear transformations. Define a linear transformation and study properties that follow from this definition, especially as they relate to matrices. Close by exploring advanced computer graphic techniques for dealing with perspective in images. x
  • 6
    Systems of Linear Equations
    One powerful application of linear algebra is for solving systems of linear equations, which arise in many different disciplines. One example: balancing chemical equations. Study the general features of any system of linear equations, then focus on the Gaussian elimination method of solution, named after the German mathematician Carl Friedrich Gauss, but also discovered in ancient China. x
  • 7
    Reduced Row Echelon Form
    Consider how signals from four GPS satellites can be used to calculate a phone's location, given the positions of the satellites and the times for the four signals to reach the phone. In the process, discover a systematic way to use row operations to put a matrix into reduced row echelon form, a special form that lets you solve any system of linear equations, and tells you a lot about the solutions. x
  • 8
    Span and Linear Dependence
    Determine whether eggs and oatmeal alone can satisfy goals for obtaining three types of nutrients. Learn about the span of a set of vectors, which is the set of all linear combination of those vectors; and linear dependence, where one vector can be written as a linear combination of two others. Along the way, develop your intuition for seeing possible solutions to problems in linear algebra. x
  • 9
    Subspaces: Special Subsets to Look For
    Delve into special subspaces of a matrix: the null space, row space, and column space. Use these to understand the economics of making croissants and donuts for a specified price, drawing on three ingredients with changing costs. As in the previous lecture, move back and forth between a matrix equation, a system of equations, and a vector equation, which all represent the same thing. x
  • 10
    Bases: Basic Building Blocks
    Using the example of digital compression of images, explore the basis of a vector space. This is a subset of vectors that, in the case of compression formats like JPEG, preserve crucial information while dispensing with extraneous data. Discover how to find a basis for a column space, row space, and null space. Also make geometric observations about these important structures. x
  • 11
    Invertible Matrices: Undoing What You Did
    Now turn to engineering, a fertile field for linear algebra. Put yourself in the shoes of a bridge designer, faced with determining the maximum force that a bridge can take for a given deflection vector. This involves the inverse of a matrix. Explore techniques for determining if an inverse matrix exists and then calculating it. Also learn proofs about properties of matrices and their inverses. x
  • 12
    The Invertible Matrix Theorem
    Use linear algebra to analyze one of the games on the popular electronic toy Merlin from the 1970s. This leads you deeper into the nature of the inverse of a matrix, showing why invertibility is such an important idea. Learn about the fundamental theorem of invertible matrices, which provides a key to understanding properties you can infer from matrices that either have or don't have an inverse. x
  • 13
    Determinants: Numbers That Say a Lot
    Study the determinant-the factor by which a region of space increases or decreases after a matrix transformation. If the determinant is negative, then the space has been mirror-reversed. Probe other properties of the determinant, including its use in multivariable calculus for computing the volume of a parallelepiped, which is a three-dimensional figure whose faces are parallelograms. x
  • 14
    Eigenstuff: Revealing Hidden Structure
    Dive into eigenvectors, which are a special class of vectors that don't change direction under a given linear transformation. The scaling factor of an eigenvector is the eigenvalue. These seemingly incidental properties turn out to be of enormous importance in linear algebra. Get started with eigenstuff" by pondering a problem in population modeling, featuring foxes and their prey, rabbits." x
  • 15
    Eigenvectors and Eigenvalues: Geometry
    Continue your study from the previous lecture by exploring the geometric properties of eigenvectors and eigenvalues, gaining an intuitive sense of the hidden structure they reveal. Learn how to calculate eigenvalues and eigenvectors; and for vectors that are not eigenvectors, discover that if you have a basis of eigenvectors, then it's easy to see how a transformation moves every other point. x
  • 16
    In this third lecture on eigenvectors, examine conditions under which a change in basis results in a basis of eigenvectors, which makes computation with matrices very easy. Discover the property called diagonalizability, and prove that being diagonalizable is the equivalent to having a basis of eigenvectors. Also explore the connection between the eigenvalues of a matrix and its determinant. x
  • 17
    Population Dynamics: Foxes and Rabbits
    Return to the problem of modeling the population dynamics of foxes and rabbits from Lecture 14, drawing on your knowledge of eigenvectors to analyze different scenarios. First, express the predation relationship in matrix notation. Then, experiment with different values for the predation factor, looking for the optimum ratio of foxes to rabbits to ensure that both populations remain stable. x
  • 18
    Differential Equations: New Applications
    Professor Su walks you through the application of matrices in differential equations, assuming for just this lecture that you know a little calculus. The first problem involves the population ratios of rats and mice. Next, investigate the motion of a spring, using linear algebra to simplify second order differential equations into first order differential equations-a handy simplification. x
  • 19
    Orthogonality: Squaring Things Up
    In mathematics, orthogonal" means at right angles. Difficult operations become simpler when orthogonal vectors are involved. Learn how to determine if a matrix is orthogonal and survey the properties that result. Among these, an orthogonal transformation preserves dot products and also angles and lengths. Also, study the Gram-Schmidt process for producing orthogonal vectors." x
  • 20
    Markov Chains: Hopping Around
    The algorithm for the Google search engine is based on viewing websurfing as a Markov chain. So are speech-recognition programs, models for predicting genetic drift, and many other data structures. Investigate this practical tool, which employs probabilistic rules to advance from one state to the next. Find that Markov chains converge on at least one steady-state vector, an eigenvector. x
  • 21
    Multivariable Calculus: Derivative Matrix
    Discover that linear algebra plays a key role in multivariable calculus, also called vector calculus. For those new to calculus, Professor Su covers essential concepts. Then, he shows how multivariable functions can be translated into linear transformations, which you have been studying since the beginning. See how other ideas in multivariable calculus also fall into place, thanks to linear algebra. x
  • 22
    Multilinear Regression: Least Squares
    Witness the wizardry of linear algebra for finding a best-fitting line or best-fitting linear model for data-a problem that arises whenever information is being analyzed. The methods include multiple linear regression and least squares approximation, and can also be used to reverse-engineer an unknown formula that has been applied to data, such as U.S. News and World Report's college rankings. x
  • 23
    Singular Value Decomposition: So Cool
    Next time you respond to a movie, music, or other online recommendation, think of the singular value decomposition (SVD), which is a matrix factorization method used to match your known preferences to similar products. Learn how SVD works, how to compute it, and how its ability to identify relevant attributes makes it an effective data compression tool for subtracting unimportant information. x
  • 24
    General Vector Spaces: More to Explore
    Finish the course by seeing how linear algebra applies more generally than just to vectors in the real coordinate space of n dimensions, which is what you have studied so far. Discover that Fibonacci sequences, with their many applications, can be treated as vector spaces, as can Fourier series, used in waveform analysis. Truly, linear algebra pops up in the most unexpected places! x

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Your professor

Francis Su

About Your Professor

Francis Su, PhD
Harvey Mudd College
Francis Su is the Benediktsson-Karwa Professor of Mathematics at Harvey Mudd College. He earned his Ph.D. from Harvard University, and he has held visiting professorships at Cornell University and the Mathematical Sciences Research Institute in Berkeley, California. In 2015 and 2016, he served as president of the Mathematical Association of America (MAA). Professor Su’s research focuses on geometric and topological...
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Mastering Linear Algebra: An Introduction with Applications is rated 4.4 out of 5 by 47.
Rated 5 out of 5 by from Understanding Vectors and Matrices ! The course gives one a very good understanding of eigenvectors and matrices. I would recommend this course to people that love mathematics.
Date published: 2020-11-16
Rated 4 out of 5 by from Fasten Your Seatbelt and Stay Alert To begin, both the course description and Dr. Su play down the necessary background in math to be successful in taking this course, saying that some calculus would be helpful, but not necessary. While that may be technically true, both the pace that the course delivers information and concepts, and the assumption by Professor Su as to the level of knowledge of his students is of a fairly high level. For example he frequently uses terms, without explanation, in discussing concepts that require a familiarity with math at a reasonable level. It is hard for me to imagine that most persons, if not all, who are taking this course will have had at least a course or two of calculus. At the least in TTC terms, I would suggest the trig and pre-calc class as well as one of the calculus ones. Perhaps it is only me, but after the first lecture, I went back and reviewed matrix math, as I felt that I was really not prepared for the speed at which the information was delivered, as well as the assumed knowledge of these skills by the professor. Not that this is a bad thing. I often feel that many of the courses offered by TTC are a little too basic and/or do not assume enough background of those taking the courses and therefore spend far too much time in delivering elementary material. There are exceptions to this general statement, the course on thermodynamics and a couple of the philosophy courses are examples requiring considerable background. Dr. Su has a low-key style, only occasionally giving us a bit of humor to leaven the material. He is soft, but well spoken, being precise, but not spending any time explaining things he thinks that we should know. This is all to the good. Where he and the course fall down is trying to cover too much material in too short a time. At least for me, I would have benefited more from the same material over 36 lectures instead of 24. This would allow for a few more examples and a bit more explanation. OTOH, for those who skills are not so ancient as mine, the pace is likely right. In any case I applaud TTC for presenting this course. I am aware that the target audience is pretty limited, but it is a gem for those of us who love math. Recommended if you are one, but if you are not, give it a pass.
Date published: 2020-11-05
Rated 5 out of 5 by from Helps understand the meaning - I've needed this I am glad I found this. Finally Linear Algebra is making some sense! I'm really only near the beginning, not close to done, but I am already thrilled that I am getting some meaning out of linear algebra instead of just performing algorithms and memorizing terms and definitions. I am doing self study and if you are too, you will also need a regular book or course with lots of exercises, because this course focuses on meaning and understanding. I should add, I have been looking at multiple sources both online and in books regarding linear algebra, and it may be that it is all of the sources together that is finally making linear algebra make sense. For anyone hesitating, I should point out the reason I am doing this is because every time I try to self study a "more interesting" higher level math subject, the math seems to rely on knowing linear algebra well first. I kept putting it off because linear algebra seemed relatively boring. The reason it was boring is because I was not really getting the meaning, not getting a deep understanding of what it all meant. I'm going for it now and am glad I did and this course really helps. I should note - they ask in the review what level of prior knowledge I had before this course and they only have 3 options. I would say I am more than a beginner but not by much - not quite intermediate. I would also like to point out I am reading the book as much as I am watching the video. I like the book because I can read my favorite parts over easily and mark pages. I like having both.
Date published: 2020-09-12
Rated 5 out of 5 by from Excellent Course! This course was a recent purchase for me. I preceded it with the Understanding Calculus I & II Great Courses. One could take the video lectures on its own and would benefit from it. On the other hand I wanted to gain a solid understanding of the Course. Just like any university course, I did the homework provided with the course. It was very beneficial and the problems really helped. I purchased the text book "Linear Algebra and its Applications, 5th Edition". It was a great supplement to the guide book and provided additional exercises. To further apply my knowledge I used a Wolfram Alpha subscription to reproduce examples in the lectures, as well as an aid in the home work. Just like college, instruction, textbook, homework and lab leads to a full understanding.
Date published: 2020-09-02
Rated 4 out of 5 by from Thought provoking Introduction. The good..... The basics of Linear Algebra presented are very good. The presented applications are very good. The overall course is a great refresher and stirs ones curiosity. The bad.... This is really only an Introduction (Hardly Masting LA). As an introduction it has far too few lectures and insufficient exercises/details. On the whole, informative and enjoyable.
Date published: 2020-06-25
Rated 4 out of 5 by from Basics & Beyond... I have a BS in Applied Mathematics and always wanted a better understanding of Linear Algebra. I particularly liked the treatment of Eigenvalues. This course had a good balance of concepts and mechanics. I sometimes wished that Prof. Su would do an example and then develop the theory. Overall a good course if you have some mathematics background.
Date published: 2020-06-01
Rated 5 out of 5 by from Interesting coverage of linear algebra, matrises a Professor Francis Su is doing a fantastic job of explaining linear equations and covers the both the algebra and geometric viewpoints. He starts out easy and early get into the matrix formulation. Toward the end the subject is a bit complicated but explained in a fashion that one follow it.
Date published: 2020-04-29
Rated 5 out of 5 by from Great application examples I had lots of exposure to linear algebra in my early years of university, but did not fully appreciate the concepts of eigenvectors, diagonalization, etc.. I was overwhelmed I guess by the abstractness and did not focus on why the theory existed. Your approach is great to get across where to apply them and to appreciate the simplification that it can bring to things like linear regression. I appreciate it more today when looking at underlying statistical models that I use in big data analytics. Thanks for the effort in producing this course..
Date published: 2020-04-04
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