3 – Linear Transformations 3

Let’s see what this looks like in the most general case, where your matrix has entries a, b, c, d. And remember, this matrix is just a way of packaging the information needed to describe a linear transformation. Always remember to interpret that first column, a, c, as the place where the first basis vector … Read more

2 – Linear Transformations 2

So how do you think you could describe these transformations numerically, if you’re say, programming some animations to make a video teaching the topic? What formula do you give the computer, so that if you give it the coordinates of a vector, it can give you the coordinates of where that vector lands? It turns … Read more

1 – Linear Transformations 1

Hey everyone, if I had to choose just one topic that makes all of the others in linear algebra start to click and which too often goes unlearned the first time a student takes linear algebra, it will be this one. The idea of a linear transformation, and its relation to matrices. For this video, … Read more

2 – Linear Combinations 2

This is a good time to talk about how people commonly think about vectors as points. It gets really crowded to think about a whole collection of vectors, sitting on a line. And more crowded still, to think about all two-dimensional vectors all at once filling up the plane. So when dealing with collections of … Read more

1 – Linear Combinations 1

In the last video, along with the ideas of vector addition and scalar multiplication, I described vector coordinates, where there’s this back and forth between, for example, pairs of numbers and two-dimensional vectors. Now, I imagined the vector coordinates were already familiar to a lot of you, but there’s another kind of interesting way to … Read more

3 – Vectors 3

All right. So, back to vector addition and multiplication by numbers. After all, every topic in linear algebra is going to center around these two operations. Luckily, each one’s pretty straightforward to define. Let’s say we have two vectors, one pointing up and a little to the right and the other one pointing right and … Read more

2 – Vectors 2

Focusing our attention on two dimensions for the moment, you have a horizontal line called the x-axis and a vertical line called the y-axis. The place where they intersect is called the origin, which you should think of as the center of space and the root of all vectors. After choosing an arbitrary length represent … Read more

1 – Vectors 1

The fundamental root of it all building block for linear algebra is the vector. So, it’s worth making sure that we’re all on the same page about what exactly a vector is. You see, broadly speaking, there are three distinct but related ideas about vectors, which I’ll call the physics student perspective, the computer science … Read more

1 – Essence Of Linear Algebra Intro

Hey everyone! So, I’m pretty excited about the next sequence of videos that I’m doing. They’ll be about linear algebra, which as a lot of you know is one of those subjects that’s required knowledge for just about any technical discipline, but it’s also I’ve noticed generally poorly understood by students taking it for the … Read more