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• "The introduction of numbers as coordinates is an act of violence."

• â€”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.

• 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

• 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

• 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 arrowâ€”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 vectorâ€”at the originâ€”to

• its tip.

• The first number tells you how far to walk along the x-axisâ€”positive numbers indicating

• rightward

• motion, negative numbers indicating leftward motionâ€”and the second number tell you how

• far to walk

• parallel to the y-axis after thatâ€”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?â€”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 movementâ€”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.

• 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 thisâ€”scaling some

• vectorâ€”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