Mastering Linear Algebra: An Introduction with Applications [TTC Video]
26 May 2019, 01:17
Course No 1056 | MP4, AVC, 2000 kbps, 1280x720 | AAC, 192 kbps, 2 Ch | 24x30 mins | + PDF Guidebook | 11.2GB
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,
- 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.
Language Families of the World [TTC Video]
11 May 2019, 21:25
Course No 2235 | MP4, AVC, 1372 kbps, 960x540 | AAC, 126 kbps, 2 Ch | 34x30 mins | + PDF Guidebook | 10.14GB
Language, in its seemingly infinite variety, tells us who we are and where we come from. Many linguists believe that all of the world’s languages—over 7,000 currently—emerged from a single, prehistoric source. While experts have not yet been able to reproduce this proto-language, most of the world’s current languages can be traced to various language families that have branched and divided, spreading across the globe with migrating humans and evolving over time.
In Language Families of the World, Professor John McWhorter of Columbia University takes you back through time and around the world, following the linguistic trails left by generations of humans that lead back to the beginnings of language. Utilizing historical theories and cutting-edge research, these 34 astonishing lectures will introduce you to the major language families of the world and their many offspring, including a variety of languages that are no longer spoken but provide vital links between past and present.
An Incomplete Family Tree
The English language comes from the immense family known as Indo-European, a group that has been traced and reconstructed perhaps most thoroughly of all the language families. In fact, it is the extensive study of this family that essentially built the foundations of formal linguistic science. Other language families, like the Niger-Congo, the Afro-Asiatic, and Austronesian families, are becoming more and more known through study, but there is still a long way to go to uncover the earliest foundations of the families that comprise the thousands of languages spoken around the world today.
Professor McWhorter demonstrates how, through a combination of the known and the unknown, of tangible evidence and shifting hypotheses, linguists trace and reconstruct languages. It’s often a tangled and complex undertaking, with many theories taking root before being reevaluated—or disproven altogether. As you better understand the methods linguists use and the ideas they have developed, you will explore a host of fascinating questions, including:
- How are similarities in languages determined?
- Why do some languages seem related but are not, while others that appear fundamentally different are actually part of the same family?
- What is the effect of geography—and even topography—on language?
- Who determines the difference between a language and a dialect?
- When does a language “officially” split into separate ones?
Filling in the Blanks of Language
As in life, the one constant in language is change. Even looking back just some hundreds of years, what we know as Middle English is barely intelligible to contemporary English speakers. Thanks to many similarities and the volume of writing that exists between the days of Chaucer’s English and now, the transition can be fairly easy to trace. However, since not every language has a clear, uninterrupted line of progression or a written record to follow through the ages, how do linguists reconstruct older languages? How do they identify a language family?
As Professor McWhorter explains: “The fundamental trait of a language family is that linguists can posit a proto-language from which the modern languages developed via regular sound changes.” This is easiest to do with groups of languages that are relatively new and thus still share a lot of features. Professor McWhorter uses the languages of Polynesia to illustrate this kind of reconstruction in its simplest form before turning to the more complicated ways linguists fill in the blanks with languages that have changed over longer periods and spread over vast distances.
Sometimes, as with the Indo-European family, there are copious written records to help cover the gaps, but often it is a matter of using core words and cognates to make the necessary connections. Like detectives, linguists must follow the clues they are given and throughout these lectures you will be able to follow the process like Watson to Professor McWhorter’s Sherlock Holmes. Along the way, you will look at language through many linguistic lenses, such as:
- Structure and parts of words, like roots, stems, prefixes, and suffixes (morphology);
- How sounds are organized in language (phonology);
- The history and origin of particular words (etymology);
- Word order and arrangement (syntax);
- The meaning and implications of words (semantics), and many more.
If language change makes it so difficult for linguists to make clear connections between past and present, it is important to understand the nature of those changes, as well as how those changes both help and hinder investigation. Languages experience change for many reasons, including:
- Time. Every generation alters the language(s) they inherit, through both the addition of new words and structures and the gradual erosion and extinction of others as cultures and societies change.
- Distance. The farther away groups of speakers become, the more linguistic changes crop up between their “versions” of the language. Sometimes this results in dialects, other times in completely new languages.
- Contact. Two unrelated languages thrown into proximity will sometimes create a mix of the two and can evolve into a new language altogether, or the influence of a dominant language can create a linguistic area with many shared characteristics among several languages.
- Force. Sometimes—often as the result of war, colonialism, or invasion—languages can be forced to change to fit a new reality or go extinct altogether.
Languages Past, Present, and Future
Languages like Chinese, Spanish, English, Arabic, Hindi, and Russian are some of the most widely spoken in the world and have been extensively studied. They can all provide deep insight into the nature of language and how it can change over time. Yet they are only a very small fraction of the immense number of languages and dialects you will encounter as you tour the world via linguistics. Following the trails of language across land and sea with Professor McWhorter will allow you to trace migration patterns and social contact between different peoples, as well as better understand important aspects of history and geography that continue to evolve and influence the world we live in today.
Utilizing maps, graphics, photographs, and a plethora of written examples and illustrations, Language Families of the World makes the complex and ever-changing world of language an engaging journey. From the “click” languages of sub-Saharan Africa and the little-known languages of New Guinea to the shrinking varieties of Native American grammar and the isolated Basque tongue in the heart of Europe, you will encounter an astonishing range of languages. Through them, you will reveal amazing facets of speech that defy conventional wisdom and demonstrate the immense range of human linguistic ingenuity.
While most animals communicate in some form, language—complete with grammar, syntax, dialects, vocabulary, and so much more—appears to be a uniquely human trait. When we understand not just the nuts and bolts but the extensive history and cultural power of language, we better understand ourselves, as well as the world and the people we share it with.
Understanding the Quantum World [TTC Video]
05 May 2019, 05:40
Course No 9750 | MP4, AVC, 2000 kbps, 1280x720 | AAC, 192 kbps, 2 Ch | 24x30 mins | + PDF Guidebook | 11.17GB
Quantum mechanics has a reputation for being so complex that the word “quantum” has become a popular label for anything mystical or unfathomable. In fact, quantum mechanics is one of the most successful theories of reality yet discovered, explaining everything from the stability of atoms to the glow of neon lights, from the flow of electricity in metals to the workings of the human eye.
At the same time, quantum mechanics does have a mysterious side, symbolized by the famous thought experiment concerning the fate of Schrödinger’s cat, a hypothetical feline who is both dead and alive in a quantum experiment proposed by Austrian physicist Erwin Schrödinger.
In Understanding the Quantum World, Professor Erica W. Carlson of Purdue University guides you through this fascinating subject, explaining the principles and paradoxes of quantum mechanics with exceptional rigor and clarity—and using minimal mathematics. The winner of multiple teaching awards, Professor Carlson is renowned for her “fantastic ability to develop and implement tools that help students learn a challenging subject”—in the words of one of her admiring colleagues. With her guidance, anyone can get a fundamental understanding of this wide-ranging field.
In these 24 half-hour lectures, you discover:
- What distinguishes quantum physics from classical physics,
- The major breakthroughs in the field and who made them,
- How to see quantum “weirdness” as a normal aspect of matter,
- Experiments that demonstrate quantum phenomena,
- Quantum theory’s many applications and physical insights,
- The probable fate of Schrödinger’s cat, and much more.
How to Learn Quantum Physics
Custom animations and graphics, analogies, demonstrations—whatever works to convey a concept, Professor Carlson uses it. You will begin Understanding the Quantum World by covering the central paradox of the field: the wave-particle duality of matter. One of the key ideas here is that waves can come in countable “quantum” units. Dr. Carlson demonstrates this with a slinky being oscillated back and forth, which generates standing waves that can be likened to quantum waves of electrons orbiting the nucleus of an atom.
Professor Carlson has a special affinity for analogies, and she uses them frequently, noting that while scientists prefer the precision of mathematics, for non-scientists an apt analogy is often the best route to an “aha” moment of insight. For example:
- The Copenhagen coin: A spinning coin is neither heads nor tails until an observation is made. Similarly, the Copenhagen interpretation considers a quantum particle to lack definitive properties until it is measured. Before that, it’s a matter of probabilities, just as a spinning coin can be considered 50 percent heads and 50 percent tails.
- Quantum gear shifter: Energy levels in an atom are quantized like the gear shifter in a car, which can go from first to second to third gear, but not to second-and-a-half. For gears, the limitation is the individual teeth in a gear wheel, while atoms are limited by the possible standing wave patterns in different atomic energy states.
- The roller coaster that could: The uncanny ability of quantum particles to pass through potential energy barriers is like a roller coaster that doesn’t have enough speed to surmount a high hill but nonetheless appears on the other side. If a coaster had a long tail to its wavefunction, then it could!
- Surfing electrons: Next time you turn on a light, think of the electrons in the wire as surfing on quantum waves, from the outer shell of one metal atom to the next, to carry current to the light bulb. Imperfections in the metal’s atomic lattice and other factors cause occasional “wipeouts,” giving rise to electrical resistance.
One of the hardest things to picture in the quantum world is the three-dimensional shape of atomic orbitals. These shapes reveal how electrons are bound to atoms and the probability of finding electrons in specific regions. Here, Dr. Carlson draws on the visualization software that physicists themselves use, which turns atoms into multicolored animations where the probability distribution is a gauzy cloud and shifting colors signify properties such as phase. These visualizations give an eerie look into a domain trillions of times smaller than the period at the end of this sentence. And for anyone studying physics or chemistry, Professor Carlson provides a handy mnemonic for remembering the nomenclature of the different atomic orbitals.
An Astonishing Range of Applications
Quantum physics is more than just a fun intellectual exercise. It is the key to countless technologies, and also helps to explain how the natural world works, including living systems. Professor Carlson discusses many such examples, among them:
- Color vision: What we perceive as color has its origin in quantum events in the outside world, which produce photons of visible light. Color-sensitive cones in our eyes detect some of these photons. Depending on their wavelength, the photons trigger quantum reactions that our brains interpret as different colors.
- Global Positioning System (GPS): GPS satellites are essentially atomic clocks in orbit, sending out very accurate time signals based on tiny transitions in energy states of cesium atoms. The time for the signal to reach Earth gives the distance to the satellite. Signals from four GPS satellites suffice to fix a position exactly.
- Flash memory: Smart phones, solid-state hard drives, memory sticks, and other electronic devices use flash memory to store data with no need for external power to preserve information. When it’s time to erase the information, quantum tunneling allows electrons that encode the data to be quickly discharged.
- Superconductivity: Dr. Carlson covers the crucial difference between the two classes of subatomic particles—fermions and bosons. Then, in a later lecture, she shows that, under special conditions, fermions can be induced to behave like bosons, leading to a frictionless state of zero electrical resistance known as superconductivity.
These and other successes in understanding and manipulating nature make the mysteries and paradoxes of quantum theory seem almost like a scientific detour into a strange new world. This is what Nobel Prize–winning physicist Richard Feynman had in mind when he urged, “I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself …‘but how can it be like that?’ because you will go … into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”
On the other hand, even as scientists invent new uses for this astonishingly powerful tool, they can’t help but speculate on how it can be like that—as you do as well in this remarkable course.