Vectors | Chapter 1, Essence of linear algebra

[Translated by Grant Sanderson. Submit corrections at criblate.com]

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 student perspective, and the mathematician's perspective.

The physics student perspective is that vectors are arrows pointing in space.

What defines a given vector is its length and the direction it's pointing,

but as long as those two facts are the same, you can move it all around,

and it's still the same vector.

Vectors that live in the flat plane are two-dimensional,

and those sitting in broader space that you and I live in are three-dimensional.

The computer science perspective is that vectors are ordered lists of numbers.

For example, let's say you were doing some analytics about house prices,

and the only features you cared about were square footage and price.

You might model each house with a pair of numbers,

the first indicating square footage and the second indicating price.

Notice the order matters here.

In the lingo, you'd be modeling houses as two-dimensional vectors,

where in this context, vector is pretty much just a fancy word for list,

and what makes it two-dimensional is the fact that the length of that list is two.

The mathematician, on the other hand, seeks to generalize both these views,

basically saying that a vector can be anything where there's a sensible notion of adding

two vectors and multiplying a vector by a number,

operations that I'll talk about later on in this video.

The details of this view are rather abstract, and I actually think it's healthy to ignore

it until the last video of this series, favoring a more concrete setting in the interim.

But the reason I bring it up here is that it hints at the fact

that the ideas of vector addition and multiplication by

numbers will play an important role throughout linear algebra.

But before I talk about those operations, let's just settle in on

a specific thought to have in mind when I say the word vector.