Video Transcript
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 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 three, negative two, 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 one,
commonly called 𝑖-hat or the unit vector in the 𝑥-direction, and the one pointing
straight up, with length one, 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 three, and
the 𝑦-coordinate as a scalar that scales 𝑗-hat, flipping it and stretching it by a
factor of two. 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 system. 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 between 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 between different coordinate
systems. But for right now, I just want you
to appreciate the fact that any time we describe vectors numerically, it depends on
an implicit choice of what basis vectors we’re using. So any time that you’re scaling two
vectors and adding them like this, it’s called a linear combination of those two
vectors.
Where does this word “linear” come
from? 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 can get, there are two things
that can happen: 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 origin. 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. Here’s some more terminology: the
set of all possible vectors that you can reach with a linear combination of a given
pair of vectors is called the span of those two vectors.
So, restating what we just saw in
this lingo, the span of most pairs of 2D vectors is all vectors of 2D space, but
when they line up, their span is all vectors whose tip sit 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 up the plane. 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 with its tail on the
origin. That way, if you want to think
about every possible vector whose tip sits on a certain line, just think about the
line itself.
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 space itself, leaving
the arrows out of it. 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 all as points. 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 space. For example, if you take two
vectors in 3D space that are not pointing in the same direction, what does it mean
to take their span? Well, their span is the collection
of all possible linear combinations of those two vectors, meaning all possible
vectors you get by scaling each of the two of them in some way, and then adding them
together.
You can kind of imagine turning two
different knobs to change the two scalars defining the linear combination, adding
the scaled vectors and following the tip of the resulting vector. That tip will trace out some kind
of flat sheet, cutting through the origin of three-dimensional space. This flat sheet is the span of the
two vectors, or more precisely, the set of all possible vectors whose tips sit on
that flat sheet is the span of your two vectors. Isn’t that a beautiful mental
image? So what happens if we add a third
vector and consider the span of all three of those guys? A linear combination of three
vectors is defined pretty much the same way as it is for two; you’ll choose three
different scalars, scale each of those vectors, and then add them all together. And again, the span of these
vectors is the set of all possible linear combinations. Two different things could happen
here: If your third vector happens to be sitting on the span of the first two, then
the span doesn’t change; you’re sort of trapped on that same flat sheet. In other words, adding a scaled
version of that third vector to the linear combination doesn’t really give you
access to any new vectors. But if you just randomly choose a
third vector, it’s almost certainly not sitting on the span of those first two. Then, since it’s pointing in a
separate direction, it unlocks access to every possible three-dimensional
vector.
One way I like to think about this
is that as you scale that new third vector, it moves around that span sheet of the
first two, sweeping it through all of space. Another way to think about it is
that you’re making full use of the three freely changing scalars that you have at
your disposal to access the full three dimensions of space. Now, in the case where the third
vector was already sitting on the span of the first two, or the case where two
vectors happen to line up, we want some terminology to describe the fact that at
least one of these vectors is redundant, not adding anything to our span. Whenever this happens, where you
have multiple vectors and you could remove one without reducing the span, the
relevant terminology is to say that they are “linearly dependent”.
Another way of phrasing that would
be to say that one of the vectors can be expressed as a linear combination of the
others since it’s already in the span of the others. On the other hand, if each vector
really does add another dimension to the span, they’re said to be “linearly
independent”. So with all of that terminology,
and hopefully with some good mental images to go with it, let me leave you with
puzzle before we go. The technical definition of a basis
of a space is a set of linearly independent vectors that span that space. Now, given how I described a basis
earlier and given your current understanding of the words “span” and “linearly
independent”, think about why this definition would make sense. In the next video, I’ll get into
matrices and transforming space. See you then!
