"The introduction of numbers as coordinates
is an act of violence." —Hermann Weyl 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 in, 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 that 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 that the order matters here. In the lingo, you'd be modelling 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 2. The mathematician, on the other hand, seeks
to generalise both of 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 that I bring it up here is
that it hints at the fact that 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". Given the geometric focus that I'm shooting
for here, whenever I introduce a new topic involving vectors, I
want you to first think about an arrow—and specifically, think about that arrow inside a coordinate
system, like the x-y plane, with its tail sitting at the origin. This is a little bit different from the physics
student perspective, where vectors can freely sit anywhere they want in space. In linear algebra, it's almost always the
case that your vector will be rooted at the origin. Then, once you understand a new concept in
the context of arrows in space, we'll translate it over to the list-of-numbers
point-of-view, which we can do by considering the coordinates of the vector. Now while I'm sure that many of you are familiar
with this coordinate system, it's worth walking through explicitly, since this is where all
of the important back-and-forth happens between the two perspectives of linear algebra. Focusing our attention on two dimensions for
the moment, you have a horizontal line, called the x-axis, and a
vertical line, called the y-axis. The place where they intersect is called the origin, which you
should think of as the center of space and the root of all vectors. After choosing an arbitrary length to represent
1, you make tick-marks on each axis to represent this distance. When I want to convey the idea of 2-D space
as a whole, which you'll see comes up a lot in these videos, I'll extend
these tick-marks to make grid-lines, but right now they'll actually get a little bit in the way. The coordinates of a vector is a pair of numbers
that basically give instructions for how to get
from the tail of that vector—at the origin—to its tip. The first number tells you how far to walk
along the x-axis—positive numbers indicating rightward motion, negative numbers indicating leftward
motion—and the second number tell you how far to walk parallel to the y-axis after that—positive
numbers indicating upward motion, and negative numbers indicating downward motion. To distinguish vectors from points, the convention
is to write this pair of numbers vertically with square brackets
around them. Every pair of numbers gives you one and only
one vector, and every vector is associated with one and only one pair of numbers. What about in three dimensions? Well, you add a third axis, called the z-axis, which is perpendicular to both the x- and
y-axes, and in this case each vector is associated with an ordered triplet of numbers: the first
tells you how far to move along the x-axis, the second tells you how far to move parallel to the
y-axis, and the third one tells you how far to then move parallel to this new z-axis. Every triplet of numbers gives you one unique
vector in space, and every vector in space gives you exactly one
triplet of numbers. So back to vector addition, and multiplication
by numbers. After all, every topic in linear algebra is going to center around these two operations. Luckily, each one is pretty straightforward
to define. Let's say we have two vectors, one pointing
up, and a little to the right, and the other one pointing right, and down a bit. To add these two vectors, move the second
one so that its tail sits at the tip of the first one; then if you draw
a new vector from the tail of the first one to where the tip of the second one now sits, that new
vector is their sum. This definition of addition, by the way, is
pretty much the only time in linear algebra where we let vectors stray away from the origin. Now why is this a reasonable thing to do?—Why
this definition of addition and not some other one? Well the way I like to think about it is that
each vector represents a certain movement—a step with a certain distance and direction in space. If you take a step along the first vector, then take a step in the direction and distance
described by the second vector, the overall effect is just the same as if you moved along the sum
of those two vectors to start with. You could think about this as an extension
of how we think about adding numbers on a number line. One way that we teach kids to think about
this, say with 2+5, is to think of moving 2 steps to the right, followed by another 5 steps to the
right. The overall effect is the same as if you just
took 7 steps to the right. In fact, let's see how vector addition looks
numerically. The first vector here has coordinates (1,2), and the second
one has coordinates (3,-1). When you take the vector sum using this tip-to-tail method, you can think
of a four-step path from the origin to the tip of the second vector: "walk 1 to the right, then
2 up, then 3 to the right, then 1 down." Re-organising these steps so that you first do all of the
rightward motion, then do all of the vertical motion, you can read it as saying, "first move 1+3
to the right, then move 2+(-1) up," so the new vector has coordinates 1+3 and 2+(-1). In general, vector addition in this list-of-numbers
conception looks like matching up their terms, and adding each
one together. The other fundamental vector operation is
multiplication by a number. Now this is best understood just by looking at a few examples. If you take the number 2, and multiply it
by a given vector, it means you stretch out that vector so that
it's 2 times as long as when you started. If you multiply that vector by, say, 1/3, it means you squish
it down so that it's 1/3 of the original length. When you multiply it by a negative number,
like -1.8, then the vector first gets flipped around, then stretched out by that factor of 1.8. This process of stretching or squishing or
sometimes reversing the direction of a vector is called "scaling", and whenever you catch a number like 2 or
1/3 or -1.8 acting like this—scaling some vector—you call it a "scalar". In fact, throughout linear algebra, one of
the main things that numbers do is scale vectors, so it's common
to use the word "scalar" pretty much interchangeably with the word "number". Numerically, stretching out a vector by a
factor of, say, 2, corresponds to multiplying each of its components by that
factor, 2, so in the conception of vectors as lists of numbers, multiplying a given vector
by a scalar means multiplying each one of those components by that scalar. You'll see in the following videos what I
mean when I say that linear algebra topics tend to revolve around these two fundamental operations: vector
addition, and scalar multiplication; and I'll talk more in the last video about how and why the
mathematician thinks only about these operations, independent and abstracted away from however
you choose to represent vectors. In truth, it doesn't matter whether you think about vectors as
fundamentally being arrows in space—like I'm suggesting you do—that happen to have a nice numerical
representation, or fundamentally as lists of numbers that happen to have a nice geometric interpretation. The usefulness of linear algebra has less
to do with either one of these views than it does with
the ability to translate back and forth between them. It gives the data analyst a nice way to conceptualise
many lists of numbers in a visual way, which can seriously clarify patterns in data,
and give a global view of what certain operations do, and on the flip side, it gives people like
physicists and computer graphics programmers a language to describe space and the manipulation of
space using numbers that can be crunched and run through a computer. When I do math-y animations, for example,
I start by thinking about what's actually going on in space, and then get the computer to represent
things numerically, thereby figuring out where to place the pixels on the screen, and doing
that usually relies on a lot of linear algebra understanding. So there are your vector basics, and in the
next video I'll start getting into some pretty neat concepts surrounding vectors, like span, bases,
and linear dependence. See you then! Captioned by Navjivan Pal
Elements of a vector space.
Unfortunately for physics students, the “computer science student’s perspective” (which is mostly promoted by mathematicians and mathematics teachers, frankly) has infected most physics instruction, such that physics students try to solve every problem in terms of separate equations in x and y coordinates, missing obvious simplifications where their chosen coordinate system is a poor fit for the problem or where it could be solved easily in a coordinate free manner.
As a result (by the time you’ve finished 15+ years of schooling in a square grid dominated society, it’s hard to get over the bias), most higher mathematics and mathematical physics has a weird fetish for using matrices for everything (including proofs of abstract theorems), even in cases where matrices are a cumbersome and geometrically uninterpretable tool for the context. This particular video doesn’t do much to divert people toward a coordinate free conception. Focusing on writing vectors in terms of some orthonormal Cartesian grid right in the first introduction is a big red flag IMO. We’ll see if the later videos focus on the geometry or if the numbers stick around persistently.
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
http://geocalc.clas.asu.edu/pdf/MathViruses.pdf
Sorry if that sounds negative. I’m all for animations, so I still hope 3blue1brown keeps going. His explanations will surely be helpful to some students, an improvement over explanations provided in their courses.
What is that software you are using to generate the animations? Sage? IPython?
This had made me realise just how good my Linear Algebra lecturers really were.
There are several different but related things that are all called vectors.
1) An equivalence class of directed line segments.
2) An ordered n-tuple of numbers
3) an element of a particular vector space that is being discussed at the moment (what space that is, is supposed to be clear from context)
Hopefully it will be clear from context which definition of the word "vector" is being used.