When I first learned about Taylor series, I definitely didn’t appreciate how important they are. But time and time again they come up in math, physics, and many fields of engineering because they’re one of the most powerful tools that math has to offer for approximating functions. One of the first times this clicked for…

# Tag: three brown one blue

## Matrix multiplication as composition | Essence of linear algebra, chapter 4

It is my experience that proofs involving matrices can be shortened by 50% if one throws matrices out. — Emil Artin Hey everyone! Where we last left off, I showed what linear transformations look like and how to represent them using matrices. This is worth a quick recap, because it’s just really important. But of…

## Change of basis | Essence of linear algebra, chapter 13

If I have a vector sitting here in 2D space we have a standard way to describe it with coordinates. In this case, the vector has coordinates [3, 2], which means going from its tail to its tip involves moving 3 units to the right and 2 units up. Now, the more linear-algebra-oriented way to…

## Three-dimensional linear transformations | Essence of linear algebra, chapter 5

[classical music] “Lisa: Well, where’s my dad? Frink: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology that Homer Simpson has stumbled into … (dramatic pause) … the third dimension.” Hey folks I’ve got a relatively quick video for you today, just sort of a…

## Implicit differentiation, what’s going on here? | Essence of calculus, chapter 6

Let me share with you something I found particularly weird when I was a student first learning calculus. Let’s say you have a circle with radius 5 centered at the origin of the xy-coordinate plane, which is defined using the equation x2 + y2=52. That is, all points on this circle are a distance 5…

## Thinking outside the 10-dimensional box

Math is sometimes a real tease it seduces us with the beauty off reasoning geometrically in two and three dimensions Where there’s this really nice back-and-forth between pairs or triplets of numbers and spatial stuff that our visual Cortex is good at processing for example if you think about a circle with radius one centered…

## Limits, L’Hopital’s rule, and epsilon delta definitions | Essence of calculus, chapter 7

The last several videos have been about the idea of a derivative, and before moving on to integrals, I want to take some time to talk about limits. To be honest, the idea of a limit is not really anything new. If you know what the word “approach” means you pretty much already know what…

## The determinant | Essence of linear algebra, chapter 6

Hello, hello again. So, moving forward I will be assuming you have a visual understanding of linear transformations and how they’re represented with matrices the way I have been talking about in the last few videos. If you think about a couple of these linear transformations you might notice how some of them seem to…