“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

Why vectors by default aways defined in column wise…….. Could you please elaborate….?

This is a really good explanation

No way you code your whole animation O_O

This video makes numbers very interesting…

I'm going to walk into Fundamentals of Mathmatics with a youtube background on linear algebra. Sweet.

Perfect one …thank you..god bless u

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. (?)

I am a CS, Physics as well as mathematics student.

I am CONFUSED.

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.

You are a hero !

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

日本語の字幕付いてる！

ありがとうございます😊

What a wonderful video!

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!

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

It's very confusing to me 🙁

Time 6:52 it would be [x1+x2 y1+y2]

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

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.

Dude, you videos are truly awesome!

This is the only video on this channel that I understand so far XD

Your way of explaining this topic got me astonished, what a wonderful teacher you are!

Good

Arent them invisible hands that can cut someone in half.

Great video! Thanks!

Aguante Quantum Fracture

what programming language do you use for your animations?

I find the way the voice is recorded distracting. It sounds like he is speaking with a bucket on his head.

math is,somehow…logic 😀

"FIND THE COMPUTER ROOM"

شكرا على الترجمة العربية

Wow, this is masterful. Thank you very much.

I can't understand 5:24

loved it . thank you soo much for the lesson

Well explained😊

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

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

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?

Duh…

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.

YOU BETTER MAKE ME FUCKING SLEEP

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

Was Professor Strang an influence?

https://www.youtube.com/watch?v=7UJ4CFRGd-U

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.

Tensor 😷📚

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.

Grant is a direct descendant of MATH GOD.

great video. you have a minor mistake at 6:51. you wrote x1+y1 and x2+y2 instead of x1+x2 and y1+y2

Вот это объяснение!

If only I had such quality teaching in my college.

great

just passing by to say how awesome your videos are.

Thanks for your work !

I go to YW (YouTube Wikipedia) University. Tuitions pretty good but not as good for networking

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?

6:52 not so ingoring the mathematician anymore? >:-D

any good vector visualization tools ? What is the tool he was using in the end ?

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

How does this guy have such a melodious voice?

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!!!

The mistake is on 6:53. The sum-of -the-vectors formula is wrong. Ha

great……….inner vector method

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.

Upload more video on vector space please

Why order matters wrt cs

Wow, that's really amazing!

1:03

Sorry for the pedantry, but wouldn't that be a 2-tuple and not a vector?

wait a second, is that beginning a fucking reference to DUCK, YOU SUCKER?

For sure, this is one of the best explanations on linear algebra. Also the video is aesthetically pleasing 😀

Yeayyy! Python

So what does a transpose do actually. I did get any intuitive explanation on it

Can any one explain pls

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

Thank you for this information

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 🙂

I am an atheist but right now I want a God to bless this guy. Keep up the good work!

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.

Why does every book on linear algebra start with vectors?

It’s first 10 chapters should be Group theory -> Vector Space-> Linear algebra

can i translate this series in Bangla ??

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..

I wish the feeling of an axiomatic transcript was indeed less axiomatic

i just saw the same video from ted ed

this is a better video

6:52 [x1,x2] + [y1,y2] = [x1 + y1][x2 + y2]

BEFORE WATCHING THIS VIDEO MY INTERPRETATION OF VECTOR IS NUMBERS AND TWO BIG SQUARE BRACKETS

if I got a good job. I will donate you my 1st-month salary.

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?

I wish I had seen this series before my most recent exam.

Please make a video on transpose of matrix, what it means geometrically?

god, learning math is so much more enjoyable when I'm not in school

This is what I want to learn, Thankyou 🙂

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

You are my crash

https://www.math-teacher-mirjana.com/

The CS student would call it a 1 dimensional list with a count of 2

You’re a beast

can you talk like a normal human being?

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.

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

I wish this guy was my teacher since like forever ago.

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.

WHERE WERE YOU IN MY UNDERGRAD… WHERE WERE YOUUU!!!!!???

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.