Оригінальна назва
3Blue1Brown Series
Випущено
06.08.2016
Жанр
Документальний
Статус
Поновлено
Кількість сезонів
4
Кількість епізодів
38
3Blue1Brown is a channel about animating math, in all senses of the word animate.
A geometric understanding of matrices, determinants, eigen-stuffs and more.
Kicking off the linear algebra lessons, let's make sure we're all on the same page about how specifically to think about vectors in this context.
The fundamental vector concepts of span, linear combinations, linear dependence, and bases all center on one surprisingly important operation: Scaling several vectors and adding them together.
Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra.
Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.
What do 3d linear transformations look like? Having talked about the relationship between matrices and transformations in the last two videos, this one extends those same concepts to three dimensions.
The determinant of a linear transformation measures how much areas/volumes change during the transformation.
How to think about linear systems of equations geometrically. The focus here is on gaining an intuition for the concepts of inverse matrices, column space, rank and null space, but the computation of those constructs is not discussed.
Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions.
Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation.
This covers the main geometric intuition behind the 2d and 3d cross products.
For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation.
This rule seems random to many students, but it has a beautiful reason for being true.
How do you translate back and forth between coordinate systems that use different basis vectors?
A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
How to write the eigenvalues of a 2x2 matrix just by looking at it.
This is really the reason linear algebra is so powerful.
The goal here is to make calculus feel like something that you yourself could have discovered.
In this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?
A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.
A visual explanation of what the chain rule and product rule are, and why they are true.
What is e? And why are exponentials proportional to their own derivatives?
Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).
Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
What is an integral? How do you think about it?
Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.
A very quick primer on the second derivative, third derivative, etc.
Taylor polynomials are incredibly powerful for approximations, and Taylor series can give new ways to express functions.
A visual for derivatives which generalizes more nicely to topics beyond calculus.
An overview of neural networks.
Introduction to neural networks.
The goal of this video is to introduce the idea of gradient descent and to analyze a specific network.
What's actually happening to a neural network as it learns?
This one is a bit more symbol heavy, and that's actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of the series, hopefully providing some connection between that video and other texts/code that you come across later.
An overview of differential equations.
How do you study what cannot be solved?
The heat equation, as an introductory PDE.
Boundary conditions, and setup for how Fourier series are useful.
Fourier series, from the heat equation to sines to cycles.
Euler's formula intuition from relating velocities to positions.
General exponentials, love, Schrödinger, and more.
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