• Professor Francis Su, PhD

Taught by Professor Francis Su of Harvey Mudd College, this course covers the topics of a first-semester college course in linear algebra, including vector spaces, dot and cross products, matrix operations, linear transformations, determinants, eigenvectors and eigenvalues, and much more. Professor Su introduces many fascinating applications of linear algebra, from computer graphics to quantum mechanics.
View Lecture List (24)
Taught by Professor Francis Su of Harvey Mudd College, this course covers the topics of a first-semester college course in linear algebra, including vector spaces, dot and cross products, matrix operations, linear transformations, determinants, eigenvectors and eigenvalues, and much more. Professor Su introduces many fascinating applications of linear algebra, from computer graphics to quantum mechanics.
View Lecture List (24)
24 Lectures  |  Mastering Linear Algebra: An Introduction with Applications
Lecture Titles (24)
• 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
Diagonalizability
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
• Professor Peter J. Smith, J.D.

A recent addition to the traditional law school curriculum, legislation and regulation are becoming more of a mainstay in some of the country’s top law schools. Law School for Everyone: Legislation and Regulation examines everything from the nature of regulation, to the challenge of interpreting statutes based on the spirit versus the letter of the law, to the role of federal agencies in our legal system.

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A recent addition to the traditional law school curriculum, legislation and regulation are becoming more of a mainstay in some of the country’s top law schools. Law School for Everyone: Legislation and Regulation examines everything from the nature of regulation, to the challenge of interpreting statutes based on the spirit versus the letter of the law, to the role of federal agencies in our legal system.

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12 Lectures  |  Law School for Everyone: Legislation and Regulation
Lecture Titles (12)
• 1
Making Sense of Legislation and Regulation
Statutes, unlike judicial opinions, tend to be brief—yet they're packed with meaning. Using a deceptively straightforward law about the use of vehicles in a public park, get an introduction to interpreting legislation and regulation. Should a statute’s plain meaning govern? Should we rely on what can be discerned about the statute’s intent? Or should we give effect instead to what seems to be the spirit of the law? x
• 2
Regulation by Statute and by Common Law
What’s distinctive about legislation as a form of regulation? In this lecture, examine how courts have applied common-law tort and contract principles in order to regulate private behavior and choices. As you’ll discover through an in-depth look at environmental regulation and incentives for car manufacturers, things are rarely—if ever—simple. x
• 3
Compared to most other Western democracies, it's much more difficult to pass legislation in the United States at the federal level. Here, Professor Smith uses the Civil Rights Act of 1964 to unpack how the unique features of the U.S. legislative process affect judicial interpretation of statutes. x
• 4
Touchstones of Statutory Interpretation
At the heart of statutory interpretation: the ability to read a legal text. Learn to do just that by thinking about how less formal kinds of interpretation in everyday life can help you interpret legal texts, and discover how 1892's Holy Trinity Church v. United States highlights the differences and similarities between interpreting legal and non-legal texts. x
• 5
The Letter versus the Spirit of the Law
A central problem related to legislation and regulation is the famous conflict between the letter and the spirit of the law. How do we reconcile the words of a statute with the legislature's apparent purpose? Study the famous 1889 case Riggs v. Palmer and 1967's Age Discrimination in Employment Act, and join the debate for yourself. x
• 6
When Is Statutory Meaning Plain?
Consider just how robust our commitment to the plain meaning of statutes should be. Cases like Tennessee Valley Authority v. Hill and West Virginia University Hospitals v. Casey illuminate whether departures from the letter of the law in order to enforce the law's spirit should be exceptions or the rule. x
• 7
Semantic and Substantive Interpretive Rules
Focus on the “canons of construction”: the additional set of background understandings that courts rely on to interpret statutes. McBoyle v. United States, from 1931, helps you grasp the difference between “semantic” canons (generalizations about conventional English language usage) and “substantive” ones (presumptions in favor of a particular set of outcomes). x
• 8
How Do Courts Really Interpret Statutes?
Using the famous case of man charged with distributing LSD, probe whether the enterprise of statutory interpretation is hopelessly incoherent and unpredictable. Also, ponder whether it's possible to articulate a theory of statutory interpretation that explains what courts actually do to resolve disputes over the meaning of statutes. x
• 9
Federal Agencies as Regulatory Bodies
Investigate how the U.S. federal government regulates, and the relationship between this regulation and legislation. You’ll focus on how agencies such as the Federal Trade Commission “enforce” federal law—and whether we should permit Congress to give agencies in the executive branch the power to decide important questions of policy. x
• 10
Political Control of Agency Decision Making
What can Congress do when it doesn’t approve of how a federal agency exercises the power Congress gave it? With this lecture, start thinking about how regulation by federal agencies—in hot-button matters such as immigration law and trade—raises critical questions about political control and constitutionality. x
• 11
Judicial Review of Agency Rulings
In the United States, judicial review by the courts is the principal way federal agencies are kept in check. Professor Smith explains two basic forms of review the courts exercise over agency decisions: ensuring that they're procedurally sound, and ensuring they're well-reasoned and based on appropriate considerations. x
• 12
Weighing Agency Interpretations of Statutes
Examine Chevron v. NRDC, the seminal case on the weight courts should give to a federal agency’s interpretation of a federal statute. Then, take a closer look at two other cases that offer a sense of how courts approach statutory ambiguity—MCI Telecommunications v. AT&T and FDA v. Brown & Williamson Tobacco. Conclude by considering the complicated interaction among legislatures, courts, and government agencies by which U.S. law and policy are implemented. x
• Professor Eric Berger, JD

Americans wage many of today’s fiercest policy debates and culture wars as battles over constitutional meaning. In the 12 lectures of Law School for Everyone: Constitutional Law, Professor Eric Berger offers the same introduction to constitutional law he provides to his own law students. You’ll come away from this course with a better understanding of our founding document’s many nuances and complexities, and the central role it plays in shaping our way of life.

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Americans wage many of today’s fiercest policy debates and culture wars as battles over constitutional meaning. In the 12 lectures of Law School for Everyone: Constitutional Law, Professor Eric Berger offers the same introduction to constitutional law he provides to his own law students. You’ll come away from this course with a better understanding of our founding document’s many nuances and complexities, and the central role it plays in shaping our way of life.

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12 Lectures  |  Law School for Everyone: Constitutional Law
Lecture Titles (12)
• 1
Origins and Functions of the Constitution
While the U.S. Constitution left many important issues unresolved, it was clearly designed to serve several primary purposes (regardless of disagreements over how it serves those purposes). Travel back to the 18th century and investigate the origins of the founding document of the American experiment—a story of crisis, rebellion, and compromise. x
• 2
The Marshall Court and the Constitution
At the core of most issues in constitutional law is one question: Who decides? So why is it that the U.S. Supreme Court became the ultimate arbiter of constitutional questions? Explore this question by examining a pair of Chief Justice John Marshall’s famous opinions—Marbury v. Madison and McCulloch v. Maryland. x
• 3
The Scope of the Executive Power
Using the 1952 opinion of Youngstown Sheet and Tube Company v. Sawyer, probe the slippery issue of how much power the U.S. president wields under the Constitution. One nuanced perspective comes from Justice Robert H. Jackson and his theory of executive power that views presidential power primarily through the lens of Congressional action. x
• 4
Congress and the New Deal Commerce Clause
Learn how Congress’s power—as we understand it today—was shaped significantly by constitutional transformations that occurred during the 1930s. See how the Court ultimately vindicated robust Congressional powers under the Commerce Clause, and how President Franklin D. Roosevelt packed the courts with judges sympathetic to his transformative New Deal policies. x
• 5
Congress and the Commerce Clause Today
Turn now to the ways the Commerce Clause has been interpreted in the decades since the New Deal era. Discover how the Court expanded Congress's power still further in Wickard v. Filburn, and how the Court revisited the Commerce Clause in cases addressing a variety of pressing social issues, including racial segregation and affordable health care. x
• 6
Individual Liberty: Contracts and Privacy
According to Professor Berger, individual liberty is one of the most doctrinally and culturally controversial topics in constitutional law. Find out how crucial a role the 14th Amendment's Due Process Clause has played when it comes to individual rights with a look at famous cases, including Lochner v. New York and West Coast Hotel v. Parrish. x
• 7
Liberty Disputed: Abortion and Gay Rights
Court decisions on some of America's most controversial issues have relied on substantive due process. Take a closer look at how the Court confronted two of these issues: abortion (in Roe v. Wade and Planned Parenthood of Southeastern Pennsylvania v. Casey) and LGBT rights (in Lawrence v. Texas and Obergefell v. Hodges). x
• 8
Equal Protection and Civil Rights
Explore the history of the Court’s civil rights decisions—including Korematsu v. United States and Brown v. Board of Education—as a way to better understand the complex relationship between law and culture. Just how did changed attitudes about race help shape seismic changes in constitutional law? x
• 9
The Affirmative Action Conundrum
Here, Professor Berger walks you through the constitutionality of affirmative action, in which public institutions give preferences on the basis of race. Key to this insightful lecture is a look at strict scrutiny, in which the Court reviews policies extremely carefully—and ostensibly without giving the government the benefit of the doubt. x
• 10
Sex Discrimination and Women's Rights
Of all the constitutions in the West, the U.S. Constitution is the only one without a provision that explicitly declares equal rights for the sexes under the law. From Minor v. Happersett in 1875 to United States v. Virginia in 1996, discover how the courts have ruled on sex-based classifications. x
• 11
The Nature of the Judicial Power
Sometimes, the courts don't decide important issues before them. In this lecture, take a closer look at why courts quite often choose not to decide a particular case on its merits. Topics here include justiciability doctrines (court-made decisions under which courts impose limitations on their power), sovereign immunity, and official immunity. x
• 12
The Politics of Constitutional Law
While judges are not (as some people assume) politicians in robes, they are certainly not immune from political influences. Explore the role of politics in constitutional law through the high-stakes confirmation battles over judicial nominees (including the battle over the seat of the late U.S. Supreme Court Justice Antonin Scalia). x
• Qi Gong Expert Lee Holden,

If you’ve been looking for exercise that can improve your health after even the most basic practice and is accessible no matter your age or body type—you’ve found it in Master Lee Holden’s Qi Gong. The practice of Qi Gong, time-tested over thousands of years, will improve your physical fitness, free your mind, and energize your life with renewed vitality. You will feel confident every step of the way as you follow Master Holden’s guidance into a more peaceful and energized life.

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If you’ve been looking for exercise that can improve your health after even the most basic practice and is accessible no matter your age or body type—you’ve found it in Master Lee Holden’s Qi Gong. The practice of Qi Gong, time-tested over thousands of years, will improve your physical fitness, free your mind, and energize your life with renewed vitality. You will feel confident every step of the way as you follow Master Holden’s guidance into a more peaceful and energized life.

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12 Lectures  |  Qi Gong for Better Health and Wellness
Lecture Titles (12)
• 1
Introduction to Qi Gong
In this lesson, you will be introduced to the flowing movements of Qi Gong, and the deep, slow breath that powers those movements. Qi is the life force that circulates through the body, bringing vitality to every organ and structure. When qi is blocked, the bodily systems can't perform at their peak, and ill health can result. The breathing, stretching, and strengthening techniques in this lesson will help ensure a fresh supply of energy to all parts of your body. You'll learn a variety of simple movements including Shaking, Spreading the Feathers, Opening the Flow, and Centering. x
• 2
Qi Gong for Better Breathing
When we spend our whole day sitting down, slightly slumped over the desk, our life force energy can become pretty low. But when we stand up, pull our shoulders back, open our heart, and breathe deeply with awareness and purpose, we elevate the life force within us. With that goal in mind, this lesson includes the movements of Wave Breathing, Holding Up the Sky, Rooster Spreads His Feathers, Soaring Crane, and Great Bear Swims in the Ocean. At the end of this lesson, you will feel both more relaxed and more energized—almost as if the air were breathing you. x
• 3
Qi Gong Mindfulness in Motion
Mindfulness is the simple-sounding act of being aware of the present moment, and recent research has shown it to have a strong positive effect on both mental and physical health. But for most of us—whose minds tend to wander here and there, over and over—mindfulness can be a challenge. In this series of Qi Gong movements, we use the power of breath and energy circulation to bring our thoughts back to the present, again and again. Movements such as Waterways, Clearing, Tiger Claw, Fireflies in the Lantern, and Lotus help us experience the present moment, exactly as it is. x
• 4
Qi Gong for the Upper Back and Neck
We live in an age of inactivity and repetitive motion, with many of us spending hours every day hunched over our computers and smartphones—all of which causes strain in the upper back and neck. But even if we had proper support, posture, and movement throughout the day, our neck muscles must support and balance our head during every waking moment. In this lesson, you’ll use deep abdominal breathing and strong but gentle movements to help clear tension from the neck, upper back, and shoulders. Movements include Silk Reelings, Water Waves, Clearing, Spreading the Feathers, and Bear Swimming in the Ocean. x
• 5
Qi Gong for Healthy Joints
When we think of our joints, most of us think of the large joints: shoulders, hips, knees, and maybe elbows. But we actually have hundreds of joints in the body, although not all are movable (like those in the skull). The energy in our moveable joints can become stagnant over time through repetitive motion, improper posture, and stress. The simple but effective movements of Qi Gong, combined with deep breathing techniques, can bring energy and suppleness back to all our joints. In this lesson, movements include Spinal Cord Breathing, Turtle Neck, Silk Reelings, and Opening the Flow. x
• 6
Qi Gong for Arms, Wrists, and Hands
This lesson is about opening energy throughout the tendons, muscles, ligaments, and joints of the lower arms, allowing for a greater range of motion and better qi circulation down through the fingers. Although arthritis symptoms often begin in the hands, many types of exercise and movement practices ignore the need to increase strength and flexibility in the hands. In this lesson, you're encouraged to listen to the needs of your body, giving yourself as much or as little stretch as you want. Movements include Silk Reelings, White Crane Soaring, Opening the Flow, and Baby Bird Learning to Fly. x
• 7
Almost everyone has experienced headache pain at one time or another, and many of us experience headaches frequently. The most common type of headache, the tension headache, can be caused by tightened muscles in the back of the neck and the scalp. Qi Gong relieves that tension, increases circulation, and calms the mind—all working together to ease the pain, and leave you feeling refreshed and energized. The movements in this lesson include Wave Breathing, Clearing, Spreading the Feathers, Waterfall, and Pulling Down the Sky to the Six Directions. x
• 8
Qi Gong for Strong Bones
In Chinese medicine, bones are considered to be the energy storage areas of the body, representing deep vitality. They are also the source of our red blood cells, made in the bone marrow. We know that if weight-bearing exercises aren't used to stimulate the bones, our bones can become more porous as we age. This Qi Gong routine uses the force of your own body weight to strengthen your bones. In addition, these exercises will help increase the flow of your qi and your overall vitality. You'll learn movements including Knocking on the Door of Life, Shaking, The Fountain, Opening the Flow, Flying, and Bone Breathing. x
• 9
Qi Gong for High Blood Pressure
When your blood pressure is consistently too high, the heart has to work harder than normal and the vessels can become damaged, resulting in a heart attack or stroke. A recent meta-analysis of studies of more than 2,000 patients concluded that Qi Gong is an effective therapy for hypertension. By using Qi Gong to relieve stress, relax your mind and body, and relieve stagnation to increase the free flow of energy, the risk of high blood pressure can be reduced. In this lesson, you will learn the movements of Water Waves, The Boa, Swimming Dragon, Repulse the Monkey, and Embracing the Tree, among others. x
• 10
Qi Gong for Energy and Vitality
Holding tension in the body—whether or not we’re aware of it—can restrict the flow of qi and dampen our sense of vitality. In this lesson, you’ll learn how to activate your own qi and how to move with strength, flexibility, and intentionality. These slow, meditative movements will strengthen the energy system of the entire body. You’ll learn movements including Knocking on the Door of Life, Chi Massage, Turning a Water Wheel, Between Heaven and Earth, Embracing the Tree, Monk Holding the Pearl, and Bamboo in the Wind. x
• 11
Qi Gong for Anxiety
While occasional anxiety is a normal and expected part of life, anxiety disorders significantly affect the lives of 40 million adults in the United States each year. The opposite of anxiety is flow, a feeling of internal peace and relaxation you can discover through the practice of Qi Gong. This lesson will help you find that equanimity, where the body is relaxed, the mind is at peace, and anxiety is held at bay. This routine is like a mini vacation from all the things in life that pull you off your center. You'll learn movements including Wave Breathing, Cleansing Breath, Drop and Shake, Qi Massage, and Cloudy Hands. x
• 12
Qi Gong Five Elements Energy Balance
There are five elements that represent the energy in nature and in each of us—water, wood, fire, earth, and metal—one energy differentiated into many forms. We ourselves are an extension of the Earth, always lifting up to the sun and the universe. The human body is a powerful conductor of electricity and qi, and when the elements and energy are balanced inside of us, we feel happy, healthy, and at peace. In this lesson, you’ll learn a variety of postures to help balance your energy, including Sword Hand, Snake Spits Out the Tongue, and Embracing the Tree. x