S2016 E1 • Fractal charm: Space filling curves

A montage of space filling curves, meant as a supplement to the Hilbert curve video.

Première diffusion : 16 janvier 2016

S2016 E2 • The Brachistochrone, with Steven Strogatz

Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.

Première diffusion : 1 avril 2016

S2016 E3 • Snell's law proof using springs

This is a supplement to the Brachistochrone video, proving Snell's law with a clever little argument by Mark Levi.

Première diffusion : 1 avril 2016

S2016 E4 • Triangle of Power

In math, exponents, logarithms, and roots all circle around the same idea, but the notation for each varies radically. The triangle of power is an alternate notation, which I find to be absolutely beautiful.

Première diffusion : 25 juin 2016

S2016 E5 • Essence of linear algebra preview

This introduces the "Essence of linear algebra" series, aimed at animating the geometric intuitions underlying many of the topics taught in a standard linear algebra course.

Première diffusion : 4 août 2016

S2016 E6 • Essence of linear algebra - Ch01 - Vectors, what even are they?

I imagine many viewers are already familiar with vectors in some context, so this video is intended both as a quick review of vector terminology, as well as a chance to make sure we're all on the same page about how specifically to think about vectors in the context of linear algebra.

Première diffusion : 5 août 2016

S2016 E7 • Essence of linear algebra - Ch02 - Linear combinations, span, and basis vectors

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.

Première diffusion : 6 août 2016

S2016 E8 • Essence of linear algebra - Ch03 - Linear transformations and matrices

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.

Première diffusion : 7 août 2016

S2016 E9 • Essence of linear algebra - Ch04 - Matrix multiplication as composition

Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.

Première diffusion : 8 août 2016

S2016 E10 • Essence of linear algebra - Footnote - Three-dimensional linear transformations

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.

Première diffusion : 9 août 2016

S2016 E11 • Essence of linear algebra - Ch05 - The determinant

The determinant of a linear transformation measures how much areas/volumes change during the transformation.

Première diffusion : 10 août 2016

S2016 E12 • Essence of linear algebra - Ch06 - Inverse matrices, column space and null space

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.

Première diffusion : 15 août 2016

S2016 E13 • Essence of linear algebra - Footnote - Nonsquare matrices as transformations between dimensions

Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions.

Première diffusion : 16 août 2016

S2016 E14 • Essence of linear algebra - Ch07 - Dot products and duality

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.

Première diffusion : 24 août 2016

S2016 E15 • Essence of linear algebra - Ch08 - Cross products

This covers the main geometric intuition behind the 2d and 3d cross products.

Première diffusion : 31 août 2016

S2016 E16 • Essence of linear algebra - Ch08 - Cross products in the light of linear transformations, part 2

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.

Première diffusion : 31 août 2016

S2016 E17 • Essence of linear algebra - Ch09 - Change of basis

How do you translate back and forth between coordinate systems that use different basis vectors?

Première diffusion : 11 septembre 2016

S2016 E18 • Essence of linear algebra - Ch10 - Eigenvectors and eigenvalues

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.

Première diffusion : 15 septembre 2016

S2016 E19 • Essence of linear algebra - Ch11 - Abstract vector spaces

The tools of linear algebra are extremely general, applying not just to the familiar vectors that we picture as arrows in space, but to all sorts of mathematical objects, like functions. This generality is captured with the notion of an abstract vector space.

Première diffusion : 24 septembre 2016

S2016 E20 • Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.

Première diffusion : 4 novembre 2016

S2016 E21 • Binary, Hanoi, and Sierpinski, part 1

Binary counting can solve the towers of Hanoi puzzle, and if this isn't surprising enough, it can lead to a method for finding a curve that fills Sierpinski's triangle (which I get to in part 2).

Première diffusion : 25 novembre 2016

S2016 E22 • Binary, Hanoi, and Sierpinski, part 2

After seeing how binary counting can solve the towers of Hanoi puzzle in the last video, here we see how ternary counting solve a constrained version of the puzzle, and how this gives a way to walk through a Sierpinski triangle graph structure.

Première diffusion : 25 novembre 2016

S2016 E23 • Visualizing the Riemann zeta function and analytic continuation

How a certain perspective on what the Riemann zeta function looks like can motivate what it might mean beyond its domain of convergence.

Première diffusion : 9 décembre 2016