Name: Linear combinations, span, and basis vectors | Essence of linear algebra, chapter 2
Uploaded: 2018-08-23T02:51:12.000Z
Duration: 9 min 59 s
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In the last video, along with the ideas of
vector addition and scalar multiplication,

where's this back and forth between, for example,
pairs of numbers and two-dimensional vectors.

Now, I imagine that 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, -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

and the one pointing straight up, with length
1, commonly called "j-hat",

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

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

In this sense, the vectors 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've chosen different basis vectors,
and gotten a completely reasonable, new coordinate

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

some other vector pointing down and to the
right, in some way.

Take a moment to think about all the different
vectors that you can get by choosing two scalars,

using each one to scale one of the vectors,
then adding together what you get.

Which two-dimensional vectors can you reach
by altering the choices of scalars?

The answer is that you can reach every possible
two-dimensional vector,

and I think it's a good puzzle to contemplate
why.

A new pair of basis vectors like this still
gives us a valid way to go back and forth

pairs of numbers and two-dimensional vectors,

but the association is definitely different
from the one that you get

using the more standard basis of i-hat and
j-hat.

This is something I'll go into much more detail
on later, describing the exact relationship

different coordinate systems, but for right
now, I just want you to appreciate the fact

any time we describe vectors numerically,
it depends on an implicit choice of what basis

So any time that you're scaling two vectors
and adding them like this,

it's called a linear combination of those
two vectors.

Why does this have anything to do with lines?

Well, this isn't the etymology, but one way
I like to think about it is that

if you fix one of those scalars, and let the
other one change its value freely,

the tip of the resulting vector draws a straight
line.

Now, if you let both scalars range freely,
and consider every possible vector that you

For most pairs of vectors, you'll be able
to reach every possible point in the plane;

every two-dimensional vector is within your
grasp.

However, in the unlucky case where your two
original vectors happen to line up,

the tip of the resulting vector is limited
to just this single line passing through the

Actually, technically there's a third possibility
too:

both your vectors could be zero, in which
case you'd just be stuck at the origin.

The set of all possible vectors that you can
reach with a linear combination of a given

So, restating what we just saw in this lingo,

the span of most pairs of 2-D vectors is all
vectors of 2-D space,

but when they line up, their span is all vectors
whose tip sits on a certain line.

Remember how I said that linear algebra revolves
around vector addition and scalar multiplication?

Well, the span of two vectors is basically
a way of asking,

"What are all the possible vectors you can
reach using only these two fundamental operations,

vector addition and scalar multiplication?"

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

So when dealing with collections of vectors
like this,

it's common to represent each one with just
a point in space.

The point at the tip of that vector, where,
as usual, I want you thinking about that vector

That way, if you want to think about every
possible vector whose tip sits on a certain

Likewise, to think about all possible two-dimensional
vectors all at once,

conceptualize each one as the point where
its tip sits.

So, in effect, what you'll be thinking about
is the infinite, flat sheet of two-dimensional

In general, if you're thinking about a vector
on its own, think of it as an arrow,

and if you're dealing with a collection of
vectors, it's convenient to think of them

So, for our span example, the span of most
pairs of vectors ends up being

the entire infinite sheet of two-dimensional
space,

but if they line up, their span is just a
line.

The idea of span gets a lot more interesting
if we start thinking about vectors in three-dimensional

For example, if you take two vectors, in 3-D
space, that are not pointing in the same direction,

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

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