In the last video, along with the ideas of
vector addition and scalar multiplication, I described vector coordinates, 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 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 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 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 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 3-D
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!.
