Placeholder Image

Subtitles section Play video

  • "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 arrowand

  • 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 vectorat the originto

  • its tip.

  • The first number tells you how far to walk along the x-axispositive numbers indicating

  • rightward

  • motion, negative numbers indicating leftward motionand the second number tell you how

  • far to walk

  • parallel to the y-axis after thatpositive 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 thisscaling some

  • vectoryou 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