Subtitles section Play video Print subtitles "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. 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 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. 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 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