Translator: Emma Gon
Reviewer: Kwangmin Lee I’m going to take you back to class, to school, to algebra, and that’s probably the last thing that you wanted to do in a TED talk today. Maybe you remember
in algebra class that you learned how to solve small systems of equations in variables X, Y, Z, and maybe they didn’t tell you exactly what these X, Y, Z were, and you were left with the impression that these equations
were not particularly interesting nor particularly beautiful. But actually equations like these are the very core
of science and engineering, and all the algebra around them. And because each of these equations defines a relationship between
these variables X, Y, and Z, and relationships
and connections are all around us. Now, for that reason, we love them, and I want to share
some of that love with you today. Maybe you remember
how to solve these equations, you express most of these variables
in terms of one or another, here Y and after some manipulations,
you get a final equation for Y that you can then solve
that gives you back X and then Z. That works really well
for these small systems of equations. But perhaps you had these nightmares of going into class one day, and your teacher writing
down on the board this test of equations and saying, “Go ahead and solve them.” Now the equations that I work with, have thousands
if not millions of variables, and obviously we would really
need a very big piece of paper, and a lot of patience to solve
in the way you were taught. So, of course, we don’t. Instead we use computer programs. Now, before we can use
these computer programs though, we need to reorder
these equations a little bit. So we’re going to write them
really really neatly under each other
and where variables are missing, for example, Z in that first equation we add it but we multiply it with a zero. And then we explicity write
all the coefficients in front of it. Write everything in terms
of addition like this. It seems a little bit silly,
why would you do that? But now you see,
every equation looks the same. Something times X
plus something times Y plus something time Z
is something else. And so we don’t have to write
X, Y and Z all the time, we just remember the order
in which they occurr, right? And we store these X, Y and Z in a little skinny table
that we call a vector, like this, and then
we store these coefficients the ones and the zeros
and here also a minus one in a separate table, like – I will show you in a minute, here. So now the system
of equations that I have is really just a table
of coefficients like this and then these little vectors
with all these unknowns. Now this table of coefficients we call the matrix. And the matrix is so famous, they even made a few movies
after it. (Laughter) And like Morpheus says in this movie, “The matrix is everywhere. It’s all around us.” Even now in this very room. So I want to give you a few examples of where matrices occur. And as first example
I am going to take you to the San Francisco Bay. So here in the San Francisco Bay, some of my colleagues designed
a really nice computer simulation of the tidal flows
going in and out the bay. And this is a really
interesting simulation that can show you for example, salinity gradients in the bay or maybe surface
velocities that are good and useful for the America’s Cup which is exactly
what they did last year. Now the flow is much
too complex to understand and computer velocity
in every single point. So instead we want to be computing the velocities in a set of points distributed throughout the domain and here the points
are the vertices of these triangles. Now through the laws of physics, we can relay the velocity
in each of these vertices to velocities in neighboring points. There is a relationship between them. If there is a relationship,
there must be an equation, and if I have a whole
bunch of these equations, what do I get? The matrix. Now, if I write down this matrix
it will be extremely large, and it will have a lot of numbers
in it, so I don’t do that, instead for every non-zero
in this matrix I put a little blue dot, right? And then what I get
is a matrix like this that we can actually look at from afar and then you can see structure in this. Now we use computer programs
to create matrices like this or visuals of matrices like this and these are called spy plots, and sometimes
these computer programs they have little Easter eggs in them. And so, one of these programs when you type the command, spy you see this. (Laughter) Now we can do the same thing in your body I don’t know if you realize this, but your body has matrices in [it]. And now I’m going to take you
to a simulation of my colleagues in Med USC in San Diego of blood flow through the aorta. Now this blood flow model was created
using CT scanning of the aorta itself. And besides the blood flow it will also give you really interesting
information about wall shear stresses that are important for blood clotting that you want to know
about for bypass operations. And again, the velocities
and the wall shear stress are computed in points
distributed like this. And the matrix that comes out has a really interesting structure, too. Now, search engines
use matrices that tell them which words occur in which website. And I want to take four
randomly chosen words: Stanford, beating, California, and the word, egg, right? And five websites that have one
or more of these words in them. And so, there is
a website over cooking, over TEDxStanford,
the Berkeley website, Now what the search engine does it creates a beautiful table, where each row of the table
corresponds to a word, and each column of the table
corresponds to a web page. And if a word occurs in the web page
then you put a little one and if it doesn’t, then you put a zero And what do you get out of this?
A matrix. Now in reality these matrices
are billions of web pages long and millions of words thick. And you need to be
a pretty good mathematician in order to build
a good search engine. So, you know now
that this is a good field to be in to earn a little bit of money. So here are those matrices. Now a very variation on this theme is shown in the next slide of a matrix and here every column, every row
corresponds to a text document, and there is a little blue dot
if these two documents have a lot of words in common. So this is a connection matrix for
20 000 documents from the Classics. We can do the same with web pages, we can put web pages
both along the rows and the columns, and put a little blue dot if there is
a hyperlink from one web page to another. And what you are looking at here now is the Stanford and Berkeley web domains. With Stanford nicely clustered,
Berkeley nicely clustered, but there is communication
between these two, because there are some blue dots
in the other side as well, which is actually kind of surprising since we keep beating them.
(Laughter) But they still want
to communicate with us. Now, matrices are also in your brain. Here you are looking
at white matter fibers that connect gray matter
regions in your brain. And looking at this,
I could create a matrix that has gray matter
regions in your brain along the rows and the columns and show connectivity
between these regions. So, here you can stare
this matrix for a long time and understand how
your brain is all connected up. So matrices are everywhere, they model systems of equations, they’re in your brain,
they’re in the search engine, and as mathematicians,
we work for them daily, and we really love them. And these matrices,
they have personalities for us. So, when I prepared for this talk, I asked my students,
what are your favorite matrices? And here are a couple
that I wanted to share with you. So we have some matrices
that are sparse with a lot of zeros, and then in the top right,
the matrices that are symmetric, and symmetry, as we know,
is always signifying beauty, so we love those. And then the matrices
that are symmetric and sparse, and sort of bended like this so they are really fantastic. And then there are other matrices
which I really like, a sort of blockage structures. But the very winner of this competition was the matrix we called
the Toeplitz matrix, and that comes up a lot
in signal processing. And it may not look
like much to you, every diagonal in this matrix
has a constant number and it makes it much easier to work with. Now what about the nastiest matrices and here is the one that we hate the most. There are two elements that are very different in size
and makes it really hard to work with. We call them ill-conditioned. But the worst matrices in the whole world are very very large matrices
that are ill-conditioned, and to be able to work with those, we need to write specialized
computer programs, and so there are millions
of lines of codes everywhere. Now the people wrote [them]
in order to manipulate these very large matrices. For example, the matrix
you heard about this morning for climate models,
it is a very nasty matrix indeed. And these lines of code, they are behind
of a lot of the simulation tools that you will see out there and that people use
in science and engineering. Now, I said, I was going to talk
about the beauty of math, and maybe so far,
you’re not super excited to say, we’ve seen some
blue dots from the screen that’s not very beautiful. So let me show you how we put beauty
into mathematics and algebra. Here is a matrix
and I’m going to associate a number with each row
and column, one through four, right? And every number is now
going to be related to a note or a little ball. There are going to be four of them,
one, two, three, four. And now whenever I have a non-zero in the matrix, for example, at positions in the first row,
positions one and three, it means there is a connection
between one and one and one and three. Now one and one doesn’t show up, but one and three does. So the next row gives me
a connection between two and four, the next row gives me an additional
connection between three and four. And now I draw
a connection, the last one, between one and four. Now we have a little graph
that comes out, and that looks much more pleasing
than just a table with numbers. But you can just imagine
that if I have a really big matrix with lots of ones and zeros
what a mess that would be. All these little balls and all
these little links between them So here is the trick. Whenever there is a link
between these two balls we imagine there is a spring
that pulls these balls together. But at the same time, we give
the balls also repelling charges, so that when they’re
not connected by a link they repel each other,
push each other away, and then we just let things go,
and sort themselves out. And what looked to be a mess,
just a simple matrix that was messy is turning into a beautiful structure. It just finds the minimum energy, and the outcome is
this fantastic geometrical figure. And when we study them,
we can actually see some of the properties of the physics, even coming out in these pictures. Now we can apply those
to lots of different matrices. And I want to share
some of the prettiest ones. Here is people
and the web pages they like. Modelling of a lung,
the matrix corresponding to that. Financial portfolio analysis. Shallow water models,
estuary flows and so on. The Standford web. MRI modelling. Analog circuits…
makes your hair stand up. Tidal flow models. And my very favorite one,
is called the galaxy that tells you how
the catalogs and sub catalogs in the Library Congress
are all connected together. So, that math that you learned
at high school that these equations
that you didn’t like so much, that’s behind everything. It is beautiful, it’s omnipresent, It’s everywhere and who knew you could make such pretty pictures. Thank you very much.
(Applause)

The beauty I see in algebra: Margot Gerritsen at TEDxStanford
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71 thoughts on “The beauty I see in algebra: Margot Gerritsen at TEDxStanford

  • June 3, 2014 at 4:25 am
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    I had already noticed stingy aspects in most financial portfolio analyses… your matrix model just confirms that… πŸ˜‰ Β 

    Magic Galaxy representation of catalogs and subcatalogs in the Library of Congress – Fascinating and inspiring talk! Thank you, Margot

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  • June 3, 2014 at 4:25 pm
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    Thanks, Roberto!

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  • July 21, 2014 at 7:47 am
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    Is there a way to find the program she uses to make matrix maps?

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  • September 8, 2014 at 10:26 pm
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    The program is called GraphViz

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  • September 10, 2014 at 7:18 am
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    Education…learning…teaching…knewing…bring good humanity for God !

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  • September 10, 2014 at 3:33 pm
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    I don't get it

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  • October 20, 2014 at 1:45 am
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    Yes yes yes . . I understand.
    Fantastic.

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  • October 26, 2014 at 4:49 am
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    Great talk, Margot!

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  • December 29, 2014 at 11:38 pm
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    An incredible, inspirational talk. This is wonderful πŸ™‚Β 

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  • January 22, 2015 at 3:35 pm
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    I love math!

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  • January 28, 2015 at 1:00 am
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    wtf so smart
    also wtf how i got here

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  • June 3, 2015 at 2:26 pm
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    The only beauty I see here is Her. Lol

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  • July 16, 2015 at 9:36 pm
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    I still use paper and board 4 and more hours per day…

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  • July 24, 2015 at 6:21 pm
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    beautiful and smart…super hard to find these days πŸ™‚

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  • July 30, 2015 at 5:54 am
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    She's so lovely, just like the Algebra she presented.

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  • December 18, 2015 at 4:12 pm
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    this is awesome

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  • January 25, 2016 at 8:35 am
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    HOLY CRAP…

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  • April 1, 2016 at 8:22 pm
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    Amazing….

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  • May 18, 2016 at 6:00 pm
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    Love your enthusiasm Margot!

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  • May 29, 2016 at 7:11 am
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    Margot makes mathematics appetizing.. Such an enthusiasm. Very rarely can lay people enjoy something a mathematician says. At least there are many physicists who know what to say to the lay people and make them understand complex phenomena. With Margot speaking, a person starts thinking to do Math even for fun as a hobby!!!

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  • June 13, 2016 at 7:40 pm
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    Cool stuff. So funny to think that back when I was in school and I was asking professors "where can I use this? and how useful it is in real life?" they did not know how to answer. Most of the professors just copy paste but have 0% application in mind.

    May be that it was a shitty school or not, the fact does not change: forcing people to memorize is the worst way of teaching. I've encountered only a few professors in my entire life and all of them online which could transcribe mathematics to real world application, how to see a problem and to translate it in mathematics thus making you understand the power of it all and why it's the basis of all sciences (my most favorite: physics).

    How can one not use mathematics? Try to solve problems efficiently by throwing banana or any other devised system/mechanism. Good luck with that, haha.

    Brilliant talk! We need more and more like her in all levels of school such that the students can correlate what they are thought with the real world and thus having more people solving problems rather than mindlessly memorizing stuff.

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  • October 5, 2016 at 9:19 pm
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    Hey Margot, echt gaaf. Uit hersenscans bij wiskundigen blijkt dat hun hersenen op exact dezelfde manier reageren bij het zien van een mooie wiskundeformule als bij een artistiek meesterwerk. Bij het bekijken van de formules worden grote delen van onze hersenen actief. . Als onze hersenen een "mooie" formule onder ogen krijgen, wordt ook het emotionele deel van onze hersenen actief. Exact op dezelfde manier als het actief wordt als we naar een mooi schilderij kijken of een mooi muziekstuk beluisteren. Hoe mooier de formule, hoe hoger de activiteit in dat precieze onderdeel van de hersenen. Dus, dank u wel. groeten vanuit Frankfurt.

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  • October 11, 2016 at 8:11 am
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    loved this

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  • October 13, 2016 at 2:42 am
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    Margot Gerritsen, you blew my mind tonight. As someone who has a BS in Mathematics and fair exposure to physics, I thought I had a good idea about the general lay of the land (I'll admit Linear algebra was not my strong point), but I'd never thought of representing systems of equations/matrices as node-edge graphs. It's beautiful! What's more, with your demonstration of an application to visualization of web page networks, I think you hinted that these same methods that you mainly applied to physical problems here are also very useful in other domains like data science/big data. The more you know. πŸ™‚

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  • October 25, 2016 at 2:12 am
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    Thank you. great presentation.

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  • October 27, 2016 at 6:38 pm
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    That"s Wonderful,Thank you so much Professor.

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  • October 29, 2016 at 7:23 am
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    So impressive : Matrices are everywhere. Thanks!

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  • November 13, 2016 at 12:12 am
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    If this isn't motivation to pay attention in linear algebra class I don't know what is.

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  • November 22, 2016 at 6:04 pm
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    i have always found algebra interesting, and was amused by how simultaneous equations solve the unknowns

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  • January 3, 2017 at 3:41 pm
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    Whoa!!

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  • January 17, 2017 at 1:33 am
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    there I was thinking it would be a talk about abstract algebra, and all I get was basic linear algebra plus programmig πŸ™

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  • February 11, 2017 at 12:15 am
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    amazing, wish I could use it in environmental protection

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  • February 25, 2017 at 2:09 pm
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    Sorry: I haven't understood a word. Not even the first explanation of how to calculate x y and z. Ms Gerritsen is addressing an audience of people who already understand algebra, and I don't. So I can't see the beauty she speaks of, which is frustrating.

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  • March 5, 2017 at 4:45 am
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    I hate maths and algebra. I quite maths when I was 15 im glad I did

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  • March 6, 2017 at 12:59 pm
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    interesting ….i really like it….

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  • March 25, 2017 at 2:10 pm
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    After years of indecision I decided to apply to a master after watching one of the professor Gerritsen talks. I cannot thank her enough for such inspirational vision about math sciences.

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  • April 3, 2017 at 8:14 pm
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    pretty interesting I guess.

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  • April 20, 2017 at 9:25 pm
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    I never saw her point of building a matrix, except maybe to demonstrate symmetry??

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  • May 12, 2017 at 8:08 pm
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    I think the honest comments are gone but they actually had good arguments and it's sad to see the same positive things about algebra and no negative things about it.
    A piano without black keys is not a piano, it's just to happy about itself and the same thing goes to this comment section. Atleast have some character

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  • May 21, 2017 at 7:02 am
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    Margot Gerritsen is truly a brilliant mathematician, and has a rare gift of being able to articulate very abstract ideas in a practical and intriguing manner. As a software developer, I long ago adopted the philosophy that computer science is about big ideas, rather than specific technologies, so I especially enjoy her lectures related to computational mathematics. She often shows how practical mathematical patterns relate to things such as search engine technologies, social networking, and more. I hope everyone else enjoys these talks as much as I do.

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  • July 9, 2017 at 7:28 am
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    blind people won't find this beuatiful

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  • August 8, 2017 at 4:55 pm
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    blah blah blah blah

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  • September 4, 2017 at 9:48 pm
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    The only problem I see is that these pictures are based on matrices that are based on samples.Β  This is like the age old debate of audiophiles about analog vs. digital recordings.Β  Analog records all frequencies and catches all of the sub-harmonics while digital recordings only sample the frequencies and don't record the full sound.Β  Margot is very intelligent and lovely and providing visual data in this format is unique, but since people look for patterns giving them only samples of the picture just doesn't seem like it would be useful to me.Β  You can digitize the Mona Lisa and then pull out sample pixels and get an entirely different picture which might have it's own beauty but how is that useful?

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  • September 15, 2017 at 5:19 am
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    This is an intriguing video and its videos like this that can make people appreciate the beauty of math. I suspect that the reason many people hate math is that they view it as intangible – that it doesn't relate to anything concrete. At least that's how it seems with the math courses I've taken. Case in point – right now I'm taking college algebra and doing exercises with compound fractions. These exercises are simply dreadful and its problems like these that kill math for many people – basically it's busy work. On the one hand those exercises are useful because they teach you precision with handling mathematical symbols as well as being able to follow mathematical rules, on the other hand they make math extremely boring as well as confusing. In my opinion these problems are soulless – these are problems that should be done by computers, not humans. Obviously math is used in engineering, chemistry, physics, economics, biology, psychology, and many other fields, the problem is they way it's taught is by not bringing those connections to light except in the occasional word problems. I can appreciate the genius of a Gauss or Euler but math as it's currently taught needs a drastic overhaul.

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  • October 1, 2017 at 11:22 am
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    Honestly marvelous 😊

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  • November 12, 2017 at 2:27 am
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    now go study 2 weeks of linear algebra and you would want to hang your self

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  • November 20, 2017 at 6:09 pm
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    Loved this

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  • December 1, 2017 at 1:36 am
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    Months ago I tried to understand this and got nowhere. I've looked at the clip once again and I can say that the lady might as well be speaking Mandarin. I don't understand ANYTHING of what she's saying.
    I wish I could find someone, somewhere, to explain what to me has always been a frustratingly closed book.

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  • December 4, 2017 at 12:35 pm
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    Another person who misunderstands and tries to deify science. We are using linear algebra to understand the world around us as a tool, not the other way around. As a mathematician, we have other tools to solve a problem just like using a duck tape or a hammer. Hammers are not omnipresent because you can break things. Linear algebra is not a good tool for every aspect about our life. It is a way for us to study flow of data or data analysis or even search in quantum physics.

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  • December 7, 2017 at 1:25 am
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    this has nothing to do with algebra, "lady"…

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  • January 7, 2018 at 4:40 pm
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    I don't get what she's talking about… matrix? Dots? Codes? Whatt? 😭😭😭😭

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  • January 17, 2018 at 8:00 pm
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    wtf this is amazing

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  • March 13, 2018 at 9:51 am
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    There is no reason to love maths this is just your illusion that you think you love maths

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  • April 4, 2018 at 12:19 pm
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    Hi Margot Gerritsen, Great talk and very inspiring. Especially the models you showed at the end. Are there any Java or python codes to develop those model based on the matrices input we give. If there are any , please share or point in the direction where I can get it . I hope many people will watch your video and get inspired with Math instead of thinking it is boring.
    Thanks again !!!!
    Regards,
    Arun

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  • April 22, 2018 at 1:40 pm
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    beauty of maths and equations

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  • May 3, 2018 at 7:18 am
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    it s really really great and interesting inspirational idea i wanna to salute you

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  • July 23, 2018 at 9:25 pm
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    I've looked again, for the third of fourth time. I still don't understand a word. So please don't say it's brilliantly elegant etc. It might be but not to people like me. Talks of vortices, matrixes… It's gibberish to me.
    The problem is that when people say they can show me how it's done, they use reasoning like this. It's useless. Then they give up.

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  • August 21, 2018 at 2:04 am
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    Anyone has idea why is it so that the square-blue-dot-pictures are symmetric along diagonals.

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  • August 24, 2018 at 1:57 pm
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    Ohh thanks dear Margot

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  • August 28, 2018 at 11:51 am
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    She is amazing!

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  • September 11, 2018 at 5:48 am
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    Mr.Baas hw πŸ‘€

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  • March 17, 2019 at 3:14 am
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    Frankly and honestly, I fail to see how algebra is anything more than solving equations.

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  • April 14, 2019 at 3:26 am
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    Linear Algebra is one of the only reasons I fire up MS Excel.

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  • April 25, 2019 at 3:20 am
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    8:54 please tell me I'm not the only one seeing the diagonal lines in the matrix? They are running through the 0.1562 numbers. Are my eyes playing tricks on me? o_0

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  • May 27, 2019 at 2:17 am
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    Algebra.., in-form-ation Calculus sum-of-all-histories function of fixed-form reciprocal numberness.., becomes the holographic matrix. Beautifully cause-effective QM-Time Principle In-form-ation.

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  • June 12, 2019 at 1:35 am
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    BAD PRESENTATION. Math is not difficult, but mathematicians are bad at presenting it.

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  • July 27, 2019 at 2:24 am
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    Superb.

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  • August 3, 2019 at 1:45 pm
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    Where are matrix in those pattern

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  • August 26, 2019 at 8:28 pm
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    It's all about relationships. Looks like STEM students aren't virgins after all.

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  • September 4, 2019 at 3:26 am
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    And none of this is taught at the correct time hahahaha

    Reply

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