it's auto did act we're back day number four of our 30 day challenge today we're getting in some actual mathematics a little bit of linear algebra i'm going to try to make a short little presentation because i got to make 30 of these i'm going to try to condense an insight i've had in the field but then we're gonna go to an old book by de morgan written in 1842 that i stumbled upon yesterday i'm gonna start reading a lot of old words in there they spelt show as shoe s-h-o-w they spelled s-h-e-w 1842 it's a long time ago alright see the library i am on a quest to understand and master high level mathematics i tried university to get this done but it wasn't the answer and so i've left it to venture out on my own to find the knowledge i desire self-educating using whatever resources i can find documenting my journey as i go in order to help you in yours the mountain is high and the path is long so let's get to climbing [Music] okay so yesterday i was looking for a book in calculus more like an older book on it and i saw de morgan which is a name i remembered from calculus class uh de morgan's law and i was like oh this is probably cool probably you reprint tell not a reprint literally this is from like 18 something when this was written in there i'm pretty sure look at that look at that that handwriting i don't even know where this book came from but like ancient were talking written by literally de morgan printed when demorgan was still kicking i was trying to find a date in it okay that says 1879 to something i think this is in a library of somewhere from 1879 to somewhere else so i was looking through it trying to find when it was printed found the prefaced preface to the second edition 1842 it's printed in 1842 i saw a handwritten quote in the beginning of it by lord calvin i don't know if you can read that lord kelvin said the reason why de morgan's great book of on the differential and integral calculus which he valued highly was now less valued than formerly was because it is too good for examination purposes life by somebody about lord calvin someone needs to say we got to read it because that's just absolutely goated [Music] presentation time so i'm gonna give a fictitious yet pertinent approach to how matrices were invented it works logically it's not how they actually came about it might be i haven't looked at the whole history but i found this a useful heuristic for learning mathematics is kind of make up some hypothetical history that's logically based and then when you go back and learn the actual history of a topic in math it's like way more of a pain in the ass like usually some genius came along and then everybody's operating here and then they went like how about this thing and then it makes absolutely no sense because they're like just stupidly smart and so the plebs have to come along like us later and then figure it out but there is usually a logical step to take and that's going to give here they were talking about matrices and what they actually are i never understood this i've taken a lot of linear algebra but never understood what we were doing matrices understood the rules of manipulation but what they actually were doing how to think of them totally had no idea and so after a lot of digging it turns out that they're actually like pretty simple uh how they evolved and there's a few logical steps you can take to get there so i'm going to take you back starting with functions if you're watching this probably what a function is single variable y goes f of x you know one number goes in one number comes out bob's your uncle so you can think of it like a machine that takes in numbers and spits out numbers there we go and then this evolves over time you know some application comes along he's like all right okay i see you i see one and one out why don't we i don't know send two numbers in why don't we send two numbers and see what happens two numbers in one number out and everyone's like i don't know dude that's kind of i mean kind of crazy dude like functions are working with the cartesian you know we were doing they were doing uh doing this thing that's represented and you have functions that's one-dimensional you know what what even is a two-dimensional function and uh this is a big probably a big leaf maybe i don't know uh two numbers in one number out that's that's the the function box now and you probably know how to represent this it's your your your or something like that so it's a you know the height that's better so this is z x y z and then represented as like height is determined by where you are on the plane and so that's all well and good then somebody else comes along and he says why don't we just make it n-dimensional why don't we have like 45 inputs to one output and uh everyone's like dude that's crazy you're crazy dog you're wild bro what are you talking about how how are we going to represent like we have three dimensions that's like a 45 dimensional function what are you talking about um and so you represent that like i don't know um y equals f of something like that and but you still have at the other day you still have one output number there's one number being output a bunch of numbers being input so then some absolute madman some bad man descends from the mountain and he goes guys i think i want to try something out i want to try something up until now you've been in one paradigm right you've been thinking one output one output number throw a bunch of stuff in but one number comes out one number output you're not thinking big enough he's like all right why don't we try hear me out here why don't we try having multiple numbers output as well and of course that caused me everyone's like you're a fool you're a charlatan throw this man in the nut house he clearly needs help but he's like hear me out here hear me out so right now he has a room full of people that's going to decide his fate he's kind of like okay guys look right now you have you know you accepted that you can have like n dimensional functions and then one output right but what if we for and let's take the simplest case what if instead of having one output how could we have it so that we have two inputs and two outputs how do we do that two inputs two outputs and so it turns out the easiest way to do this is to just say uh go like this you want two outputs right and so let's say that and we're just coming up with this arbitrary notation we're just trying to come up with something we haven't operated before we're just trying to say okay how do they represent this two-dimensional output it's a whole new thing right so he does this he comes up with uh this thing and he's like okay we're gonna put the first equation up here i'm gonna say uh x y let's say g of x y oh we're gonna come up with newton's notation because how do you put two numbers as an output to a function it's all very strange we're gonna have this as uh we're gonna call these i don't know what's the symbol i can use uh d w let's say z w yeah we can use z w or w z do you feel like you just created a new branch of mathematics because you did bam multi-dimensional outputs so we have a vector from a vector so this is is a way of representing a multi-dimensional output and input so you have two variables x and y coming in and you have two variables coming out and let's let's there's another way of representing this let's go uh because this is i mean this is i know kind of here we just invented some spot uh so over time you know we want to put it in normal function notation um and so instead of that we're going to have uh let's say just put f of f here and g of you these are functions we're going to have uh maybe put them like that i don't know kind of poor notation but we're working with the notation here so it's all good we're inventing notation um and then because we have two inputs going in we're gonna put them like this and so all this means is like go like this right so f of something and then find out whatever that is and g of this and find whatever that is um and so we're going to say equals so that's how you represent multi-dimensional outputs for multi-dimensional inputs and then hopefully you're noticing some similarity between this and matrices we're almost there so matrices are just this and and with this you can you can go i don't know you have like 20 inputs and 20 outputs or like two inputs and 14 outputs like you can make these crazy stupid but at the end of the day it's a function we're operating with functions functions think functions it's a function and so all a matrix is oh also there's a way of representing this thing um watch three blue one brown's uh linear algebra series it's like you take a so let's say it's a two dimension of two-dimensional it's a four-dimensional thing what you're operating with but you can kind of take a take the original graph superimpose a new graph on top of it and then wherever the lines go that becomes like the new point so it's like the function takes the point one one it moves it to like thirteen fourteen it'll like move it over there um you but watch three blue one browns you'll see what i'm talking about i don't have the technology at this point to do my own okay so then the only next step we're gonna do is limit this to linear equations a linear equation a linear equation is any equation that just multiplies it by some number so uh y equals sine of x not linear not linear not multiplying by some coefficient right it's just not just a nice easy y equals blank of x it's y equals sine of x it gets all curvy wrong incorrect not linear um y equals e to the x not linear unfortunately um y equals four x that's linear linear because it's a line it's a line uh and also coefficients aren't allowed i i had a big problem with this like what about y equals four x um plus two why what about that and that actually turns out not to be fitting our thing uh it's a whole i dove into this while that's the case it's very complicated but just this you don't you're not allowed to have like plus twos on it some number multiplied by the variable that's a linear function a multi-dimensional linear function is equals so it's just coefficients multiplied by the variables uh so this is a this is yep that's good if a and b are are just normal numbers um or not normal just numbers they don't need normal numbers just real numbers sorry but this this is no good that's not linear we're not that's not linear unfortunately not linear does not make a cut okay if you have linear numbers and you have multi-dimensional functions or sorry you have linear functions and multi-dimensional functions linear functions multi-dimensional functions peanut butter and chocolate time you ready bring them together what we got what do we got what we got is matrices baby when you see a matrix thick function matrices are functions they take in vectors they spit out vectors vector comes in spits out a vector so matrix is just this or some larger version of this so maybe it's like it could be three long it could be arbitrarily long and taking arbitrarily sized vectors but the other day it's just a way of doing multi-dimensional linear outputs so you take it takes a vector in it spits a vector out and it does so linearly that's the trick it's it's a it's a way of doing multi-dimensional functions and the only restriction on that is it's linear it's a linear way of doing it so that does the i erased it but it does it multiplies every variable by some number because it just does the linear thing there's an easier way of representing it uh you can just put the coefficients in the actual function notation and so we do that by like so what this says what this says that took me forever to understand this is just a function it's just a function and it takes in this this is the thing this is the argument this is the argument and this is the function this says take this right x y z take x multiply by one take y multiply two take z multiply by three same thing for this row take four multiply by y take five multiply it by vertical number take four multiply by x sorry take five multiply y take six multiply z and we just invented this we invented this notation this way of multiplying it it's just a way of doing it i mean come up with your own way of trying to take a vector as an input and have a vector as an output just see how you can do it there's a ton of ways to notify it this is just one we've come up with but it's just an arbitrary way of writing it down um and so this is a function you see a matrix think function multi-dimensional in multi-dimensional act and they don't need to be square so you could have uh you know you can have it like so this one this one would take now so this one would take two dimensional input a two dimensional vector fired into the function and it outputs a four dimensional vector uh because it has it has four different things so it'll have like i don't even know it would be so this is a four dimensional vector from a two-dimensional vector and you can reverse it you could have a four-dimensional vector come in two-dimensional vector come out you could have like a thirty-seven thousand 492 sized vector come in and like a 49 000 vector come out you get crazy with it and the study of linear algebra is a study of these functions so there's a bunch of stuff you can do it gets quirky and there's a bunch of different eigenvalues eigenvectors like determinants i hate determinants determinants are my mortal enemy not a fan of terminus but you get the point it so it's but linear algebra when you look at linear algebra is just a study of linear multi-dimensional functions you take in a vector you linearly do something to it and you fire another vector i was intending that to be like two minutes long but this is gonna be a pain in the ass to edit it needed to be said though i'm gonna do a whole series on linear algebra so uh if this got confusing when i try to do a little bit better in the future but when you see matrices what i really want you to understand from this the one main thing i want to i want to get across is when you see a matrix it's a function see a matrix as a function and that should clear up a lot of stuff it did for me anyway all right maybe i do an outro maybe i don't if i don't do an outro i love you guys take it easy and godspeed i will see you in the next one which is tomorrow boy do we have a lot to edit tonight holy cow [Music] [Applause] [Music] foreign