Vectors, what even are they? | Essence of linear algebra, chapter 1

Vectors, what even are they? | Essence of linear algebra, chapter 1

“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

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  1. The Mathematicians and Scientific others are all relying on the Physicists for the intuition of "field", "origin" and measurable dimensions that are composites of temporal resonance vectorization.
    That's derived from e-Pi-i resonance, the naturally occurring conception of synchronous location and relative numberness differential coordination, in the context of Calculus anti-log scalar probability integration. (?)

  2. Your videos are like a big recap Seminar where out of sudden everything seems so logical and makes sense. Throughout my studies I strive for interconnections between the disciplines of mathematics CS business and finance but it has never been so obvious since I came into contact with your impassionating lessons.

  3. Nice.
    One little detail : the ticks on the axes X,Y and Z are often represented in a confusing way : one can't tell quickly the difference between half ticks and integer ticks. At first sight, the first tick seems to be 1 instead of 0.5

  4. Thanks thanks thanks! I just wanted to visualize it. my imagination power collapsed trying but you video made it crystal clear! humble regards from India!

  5. you showed one vector by V and other by w why is that? m studying vectors for the first time, a little help please!!!

  6. You should recommend us some books with a similar aproach yo the subject (the same with the calculus series) If there are not any, I would really ask you to write one. It is amazing

  7. Obviously this video is more about the fundamentals, but as a nerd I'm curious how this relates to stellar navigation. Like how to compensate when coordinates are never consistent, ie. the expanding Universe, time dilation and all that other stuff I barely grasp. Specifically in Star Trek, how Picard could have all of these vectors in his head which he could just spout off at anytime without any calculations as to their current position and the movement of space. The whole TNG heading routine (883-mark-41 or whatever) never seemed adequate to me. But then again why have someone at the helm in the first place when you can just tell the computer to "take me to Vulcan" in plain speak. Like I said, nerd.

  8. lütfen devam edin … yorulduğunuzda ; insanlık için attığınız adımların sizin gibileri çoğalttığını ve bir bayrağı teslim etmenin ölmemek demek olduğunu huzurla ve yaşamsal(ölümcül) arzuyla hissedin

  9. I'm curious how this looks in Vector Synthesis when you have a sample frame added with filter and envelope and modulation…

  10. I found it odd that you used the z axis as the vertical @ 4:30, essentially teaching the x, y as left/right, up/down – only then to rotate it and make the z axis the up/down. I'm several years out of math classes, but brushing up for coding, and I thought this was odd. Is that correct?

  11. Why don't you make a video about Tensors?
    There are not many good videos in youtube about tensors in youtube which explains tensors intuitively.

  12. I count myself as one of the luckiest people to have stumbled on this video just before taking linear algebra in college

  13. Thanks for putting this much of time and effort to this. I wish I was back in school and had the opportunity to learn from scratch like that.

  14. Interesting you say the physics perspective is free floating, which backs reality…where the rest of you are using a controlled environment or closed system. Good luck finding such a thing in the real world. If you want any sort of accuracy in predictions, you need a valid zero point.

  15. Hello sir, i have two queries. will the series of videos be helpful for Data science purpose? Why do we need linear algebra in Data Science?

  16. How can someone volunteer to add captions in their local language ? This is too good to not to tell to my students here in vavilala ( a village in south India ) where subjects are taught in Telugu language

  17. It's the better video, in internet, about the vecteur, all over the world!!! And, I'm French. C'est la meilleure de toutes les vidéos sur l' Introduction aux notions de vecteurs, donc d'espace vectoriel, donc de matrice, donc de tenseur….. Extraordinaire, magnifique, exceptionnel, irremplaçable, et génial!!!

  18. I swear to god i wish my teachers could just simply fucking explain shit, let alone as well as you do id know all of this by know but never have we ever normally started thinking about what the fundamentals are in vector science but simply all we know are the surface level knowledge of it meaning we dont really understand why they work and how its actually applicable to the actual world. Thats what makes people say that mathematics are just simply throwing numbers around and that theyll never even need them so they can just simply forget what they knew at the end of high school. When someone explains all of this to you youre left with a deep enough knowledge to apply it without ever having to forget how it goes because you already know the fundamentals AND youre left with no questions regarding the use or applicability in life. Knowing these things and being this way typically tends to make someone love the subject because they start seeing it as easy because now they understand it at a higher level + the added challenge of the capable difficulty you can come over makes a person being attracted to the feeling of being accomplished when youre really good at something. Its just simply an ultimate letdown that ive not seen any single teacher in my 14 years of highschool experience that explained something to me on a level this deep since the whole ''glossing over the curricula to fit in a year'' thing made me hate most of the subjects that i find interesting and would gladly learn on a level where i could actually understand what they are and wanting to even know more.

  19. Literally can't thank you enough for making these videos. I would have failed my class and subsequently failed out of college without the understanding these vids helped me get. You're doing good work here

  20. You are genious! Why do not all my professors at the university teach like you do?! Also one thing to mention is the great and fascinating animation! Well done 🙂

  21. This video is horrible. First you change the scale of the tick marks 6:07 1 tick on x =1; 2 ticks on y =1. 6:12 ticks on x are suddenly 2 ticks =1. THEN you have the vectors with coordinates 1(x1) 2(y1) and 3(x2) and -1(y2) and you SAY you add them straight across, but your vectors say you add x1 and y,1 not x1 and x2. In other words at 6:52 your green x2 magically changes to y1.

  22. Why does every book on linear algebra start with vectors?
    It’s first 10 chapters should be Group theory -> Vector Space-> Linear algebra

  23. Omg.. Many of my questions are answered In this single video.. Thankyou so much sir for making this much effort.. God bless you…now I could visualise linear algebra.. It will be easy for my comprehension.. Thankyou so much sir..

  24. hey Grant,
    what is the relation between complex numbers and vectors? are they same? can be represented by an ordered pair, addition follows same rules.
    if they are same where does the sqrt(-1) joins in?

  25. Hi, the videos are very helpful in understanding the concept of vectors. However, Can you please suggest a good and simple book on the given chapters form 1 to 10 for easy recollection of the understood concepts

  26. Physics student from PSU here. I wanted to thank you for all the great vocal warm-ups. "Ohhh!"; "Ahhh"; "Ohhheyeseee"; "Ohhhnooowaaay"; and one more that isn't really appropriate outside the military. Or in it. Or anywhere really.

  27. in my mind I sometimes imagine myself explaining a subject perfectly, and in practice I always fall short. Your videos are what I picture a masterful presentation to look like…I wonder if my missing piece is mastery of the editing tool you are using because you seem able to put your thoughts into edited content

  28. Can you post some example code (in its entirety) that you use to generate animations? That's something I was actually curious about; I had a feeling you did it using some kind of code rather than using After Effects or something.

  29. Great video, one small thing I feel compelled to mention. At around 04:06 when you introduce the Z axis, you do so as a vertical axis. In both graphic design and game development, two fields which would use vectors and coordinates all the time, the vertical axis is always the Y axis, with the Z being the 'depth' axis. The reason for this is it allows switching between 2D and 3D perspectives with the X and Y axis remaining the same. I don't know if this is the same in mathematics.

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