Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

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 imagine the vector coordinates were already familiar to a lot of you,

but there's another kind of interesting way to think about these coordinates,

which is pretty central to linear algebra.

When you have a pair of numbers that's meant to describe a vector,

like 3, negative 2, I want you to think about each coordinate as a scalar,

meaning, think about how each one stretches or squishes vectors.

In the xy coordinate system, there are two very special vectors,

the one pointing to the right with length 1, commonly called i-hat,

or the unit vector in the x direction, and the one pointing straight up with length 1,

commonly called j-hat, or the unit vector in the y direction.

Now, think of the x coordinate of our vector as a scalar that scales i-hat,

stretching it by a factor of 3, and the y coordinate as a scalar that scales j-hat,

flipping it and stretching it by a factor of 2.

In this sense, the vector that these coordinates

describe is the sum of two scaled vectors.

That's a surprisingly important concept, this idea of adding together two scaled vectors.

Those two vectors, i-hat and j-hat, have a special name, by the way.

Together, they're called the basis of a coordinate system.

What this means, basically, is that when you think about coordinates as scalars,

the basis vectors are what those scalars actually, you know, scale.

There's also a more technical definition, but I'll get to that later.

By framing our coordinate system in terms of these two special basis vectors,

it raises a pretty interesting and subtle point.

We could have chosen different basis vectors and

gotten a completely reasonable new coordinate system.

For example, take some vector pointing up and to the right,