Kind: captions
Language: en
the following is a conversation with
Gilbert Strang he's a professor of
mathematics at MIT and perhaps one of
the most famous and impactful teachers
of math in the world his MIT
opencourseware lectures on linear
algebra have been viewed millions of
times as an undergraduate student I was
one of those millions of students
there's something inspiring about the
way he teaches there's at once calm
simple and yet full of passion for the
elegance inherent to mathematics I
remember doing the exercises in his book
introduction of linear algebra and
slowly realizing that the world of
matrices of vector spaces of
determinants and eigenvalues of
geometric transformations and matrix
decompositions reveal a set of powerful
tools in the toolbox of artificial
intelligence from signals to images from
miracle optimization to robotics
computer vision deep learning computer
graphics and everywhere outside AI
including of course a quantum mechanical
study of our universe this is the
artificial intelligence podcast if you
enjoy it subscribe on YouTube give it
five stars an apple podcast support on
patreon or simply connect with me on
Twitter Alex Friedman spelled Fri D ma N
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better world and now here's my
conversation with Gilbert Strang how
does it feel to be one of the modern-day
rock stars of mathematics I don't feel
like a rock star that's kind of crazy
for old math person but it's true that
the videos in linear algebra that I made
way back in 2000 I think I've been
watched a lot and well it's partly the
importance of linear algebra which way
I'm sure you'll ask me and give me a
chance to say that linear algebra as a
subject has just surged in importance
but also I it was a class that I taught
a bunch of times so I kind of got it
organized and any I'm enjoy doing it was
just the videos were just the class so
they're on OpenCourseWare and
YouTube and translated that's fun but
there's something about that chalkboard
in the and the simplicity of the way you
explain the basic concepts in the
beginning I you know to be honest when I
went to undergrad you know do linear
algebra broadly of course this lineage
I before going through the course at my
university I was going through open
courseware I was you were my instructor
oh yeah you're right yeah and that I
mean we're using your book and I mean
that that the fact that there is
thousands you know hundreds of thousands
millions of people that watch that video
I think that's yeah that's really
powerful so how do you think the idea of
putting lectures online would really MIT
OpenCourseWare has innovated that was a
wonderful idea
you know I think the story that I've
heard is the committee committee was
appointed by the president president
vest at that time a wonderful guy and
the idea of the committee was to figure
out how a mighty could make be like
other universities market the market
work we were doing and then they didn't
see away and after a weekend and they
had an inspiration came back to the
president vest and said what if we just
gave it away and he decided that was ok
good idea so that's a crazy idea that's
uh if we think of a university is a
thing that creates a product yes isn't
knowledge right the you know the kind of
educational knowledge isn't the products
and giving that away are you surprised
that you went through it the result that
he did it well knowing a little bit
president vest it was like him I think
and and it was really the right idea
you know MIT as I kind of it's known for
being high level technical things and
and this is the best way we can say tell
we can show
what MIT really is like because the the
in my case those 1806 videos are just
teaching the class they were there in
26100 they're kind of fun to look at
people write to me and say oh you've got
a sense of humor but I don't know where
that comes through somehow I big
friendly with a class I like students
and and then your algebra the subject we
got to give this subject most of the
credit it it really has come forward and
importance in these years so let's talk
about linear algebra a little bit
because it is such a it's both a
powerful and a beautiful a subfield of
mathematics so what's your favorite
specific topic in linear algebra or even
math in general to give a lecture on to
convey to tell a story to teach students
okay well on the teaching side so it's
not deep mathematics at all but I I'm
kind of proud of the idea of the four
subspaces there are four fundamental
subspaces
which are of course known before long
before my name for them but can you go
through them can you go through the
future I can yes so the first one to
understand is so the matrix is maybe I
should say the matrix what is the matrix
what's a matrix well so we have a like a
rectangle of numbers so it's got n
columns
got a bunch of columns and also got an M
rows let's say and the relation between
so of course the columns and the rows
it's the same numbers so there's got to
be connections there but they're not
simple the they're much the columns
might be longer than the rows and
they've all different the numbers are
mixed up first space to think about is
take the columns so those are vectors
those are points in n dimensions what's
the vector so a physicist would imagine
a vector or might imagine a vector as a
arrow you know in space or the point it
ends at in space for me it's a column of
numbers does it you often think of this
is very interesting in terms of linear
algebra of a vector you think a little
bit more abstract than the how it's very
commonly used perhaps yeah you think
this arbitrary Speight multi-dimensional
right away I'm in high dimensions and in
the room lands yeah that's right in the
lecture I tried a so if you think of two
vectors in ten dimensions I'll do this
in class and I'll readily admit that I
have no good image in my mind of a
vector of arrow int n dimensional space
but whatever you can a you can add one
bunch of ten numbers to another bunch of
ten numbers so you can add a vector to a
vector and you can multiply a vector by
three and that's if you know how to do
those
you've got linear algebra you know ten
dimensions yeah you know there's this
beautiful thing about math if we look
string theory and all these theories
which are really fundamentally derived
through math yeah but it very difficult
to visualize it yeah how do you think
about the things like a 10 dimensional
vector that we can't really visualize
yeah do you and and yet math reveals
some beauty Oh underlying me yeah our
world in that weird thing we can't
visualize how do you think about that
difference well probably I'm not a very
geometric person so I'm probably
thinking in three dimensions and the
beauty of linear algebra is that is that
it goes on to ten dimensions with no
problem I mean that if you're just
seeing what happens if you add two
vectors in 3d you then you can add them
in ten D or you're just adding the ten
components so so I I can't say that I
have a picture but yet I try to push the
class to think of a flat surface in ten
dimensions so a plane in ten dimensions
and so that's one of the spaces take all
the columns of the matrix take all their
combinations so on so much of this
column so much of this one then if you
put all those together you get some kind
of a flat surface that I call a vector
space space of vectors and and my
imagination is just seeing like a piece
of paper in 3d but anyway so that's one
of the spaces that's space number one
the column space of the matrix and then
there's the row space which is as I said
different but came came from the same
numbers so we got the column space all
combinations of the columns and then
we've got the row space all combinations
of the rows so those are those words are
easy for me to say and I can't really
draw them on a blackboard but I try with
my thick chalk everybody everybody likes
that railroad chalk and me too
I wouldn't use anything else now
and and then the other two spaces are
perpendicular to those so like if you
have a plane in 3d just a plane is just
a flat surface in 3d then perpendicular
to that plane would be a line so that
would be the null space so we've got two
we've got a column space a row space and
they're two perpendicular spaces so
those four fit together in the in a
beautiful picture of a matrix yeah yeah
it's sort of a fundamental it's not a
difficult idea comes comes pretty early
in 1806 and it's basic planes in these
multi-dimensional spaces how how
difficult of an idea is that to come to
do you think if you if you look back in
time yeah I think mathematically it
makes sense but I don't know if it's
intuitive for us to imagine just what
we're talking about
feels like calculus is easier to I see
into it well calculus I have to admit
calculus came earlier earlier than
linear algebra so Newton and Leibniz
were the great men to understand the key
ideas of calculus but linear algebra to
me is like okay it's the starting point
because it's all about flat things
calculus has got all the complications
of calculus come from the curves the
bending this is a curved surfaces linear
algebra the surfaces are all flat
nothing bends in linear algebra so it
should have come first but it didn't and
calculus also comes first in in high
school classes in in college class it'll
be freshman math I'll be calculus and
then I say enough of it like okay get to
get to the good stuff and that you think
linear algebra should come first
well it really yeah I'm okay with it not
coming first but it should yeah
it should it's simpler because
everything is flat yeah everything's
flat of course for that reason you
sort of sticks to one dimension or so or
eventually you do multivariate but that
basically means two dimensions linear
algebra you take off into ten dimensions
no problem it just feels scary and
dangerous to go beyond two dimensions
that's all if everything is flat you
can't go wrong
so what concept or theorem in linear
algebra or in math you find most
beautiful it gives you pause that leaves
you and oh well I'll stick with linear
algebra here I hope that viewer knows
that really mathematics is amazing
amazing subject and deep deep
connections between ideas that didn't
look connected some they turned out they
were but if we stick with linear algebra
so we have a matrix that's like the
basic thing a rectangle of numbers and
might be a rectangle of data you're
probably going to ask me later about
data science where and often data comes
in a matrix you have you know maybe
every column corresponds to a to a drug
in every row corresponds to a patient
and and if the patient reacted favorably
to the drug then you put up some
positive number in there anyway
rectangle of numbers a matrix is basic
so the big problem is to understand all
those numbers you got a big big set of
numbers and what are the patterns what's
going on and so one of the ways to break
down that matrix into simple pieces is
uses something called singular values
and that's come on as fundamental in the
last and certainly in my lifetime I can
values bro you if you have viewers
who've done engineering math or or more
basic linear algebra eigenvalues were in
there
but those are restricted to square
matrices and data comes in rectangular
matrices so you got to take that you got
to take that next step I'm I'm always
pushing math faculty get on do it don't
do it
do it singular values so those are a way
to break to make to find these the
important pieces of the matrix which add
up to the whole matrix so so you're
breaking a matrix into simple pieces and
the first piece is the most important
part of the data the second piece is the
second most important part and then
often so a data scientist will like if
if a data scientist can find those first
and second pieces stop there the rest of
of the data is probably round off you
know we're yeah experimental error maybe
so you're looking for the important part
yeah so what do you find beautiful about
singular values well yeah I didn't give
the theorem so here's the here's the
idea of singular values every matrix
every matrix rectangular square whatever
you can be written as a product of three
very simple special matrices so that's
the theorem every matrix can be written
as a rotation times a stretch which is
just a matrix diagonal matrix otherwise
all zeros except on the one diagonal and
then a third and the third factor is
another rotation so rotation stretch
rotation is the breakup of a of any
matrix the structure that the ability
that you can do that what do you find
appealing what do you find beautiful
bodies well geometrically as I freely
admit the mate action of a matrix it's
not so easy to visualize but everybody
can visualize a rotation take-take-take
two-dimensional space and just turn it
around the around the center
take three-dimensional space so a pilot
has to know about well what are the
three the yaw is one of them I've
forgotten all the three turns that a
pilot makes up to ten dimensions you've
got ten ways to turn but you can
visualize a rotation take the space and
turn it and you can visualize a stretch
so to break a a matrix with all those
numbers in it into something you can
visualize rotate stretch rotate it's
pretty neat pretty neat that's pretty
powerful on YouTube just consuming a
bunch of videos and just watching what
people connect with and what they really
enjoy and are inspired by math seems to
come up again and again I I'm trying to
understand why that is perhaps you can
help yeah I mean give me clues so it's
not just to let the kinds of lectures
that you give but it's also just other
folks were like with numberphile there's
a channel where they just chat about
things that are extremely complicated
actually
yeah people nevertheless connect with
them you know what do you think that is
what it's wonderful isn't it I mean I
wasn't really aware of it do so we're
we're conditioned to think math is hard
math is abstract math is just for a few
people but it isn't that way a lot of
people quite like math and they liked it
I get messages from people saying you
know now I'm retired I'm gonna learn
some more math I get a lot of those it's
really encouraging and I think what
people like is that there's some order
you know a lot of order and or you know
things are not obvious but they're true
so it's really cheering to think that
that so many people really want to learn
more about math yeah in terms of truth
again sorry to slide into philosophy at
times yeah math does reveal pretty
strongly what things are true
yeah I mean it's the whole point of
proving things is and yet sort of our
real world is messy and complicated what
do you think about the nature of truth
that math reveals oh wow because it is a
source of comfort like you've mentioned
yeah that's right
well I have to say I'm not much of a
philosopher I just like numbers you know
I think yeah I would you this was before
you had you had to go in when you're in
the other filling your teeth yeah I kind
of just take it yeah so I what I did was
think about math you know like take
powers of 2 2 4 8 16 up until the time
the two stopped hurting and the dentist
said you're through or Counting yeah so
so that was the source of just such a
piece almost yeah what what what what is
it about math you think that brings that
yeah what is that well you know where
you are yeah symmetry it's it's
certainty the fact that you know if
you'd to if you multiply 2 by itself 10
times you get a thousand 24 period
that's everybody's gonna get that do you
see math is a powerful tool or is an art
form so it's both that's that's really
one of the neat things you can you can
be an artist and and like math you can
be a engineer and use math which are you
which am I what did you connect with
most yeah I'm in here between I'm
certainly not a artist type philosopher
type person might sound that way this
morning but I'm not yeah I I really
enjoy teaching engineers because they
they they go for an answer and yeah so
of probably within the mountain MIT math
department most people enjoy teaching
people teaching students who get the
abstract idea I'm okay with with I'm
good with engineers who are looking for
a way to find answers
yeah actually that's a interesting
question do you think do you think for
teaching and in general but thinking
about new concepts do you think it's
better to plug in the numbers or to
think more abstractly so looking at
theorems and proving the theorems or
actually you know building up a
basically tuition of the theorem or the
method the approach and then just
plugging in numbers and seeing it work
you know well certainly many of us like
to see examples first we understand it
might be a pretty abstract sounding
example like a three dimensional
rotation how are you gonna how are you
gonna understand a rotation in 3d or in
10 D or but and then some of us like to
keep going with it to the point where
you got numbers where you got 10 angles
10 axes 10 angles but the best the great
mathematicians is probably I don't know
if they do that because they they for
them for them an example would be a
highly abstract thing to the rest of it
right but nevertheless working within
the space of examples yeah example it
seems to examples of structure our brain
seemed to connect with that yeah yeah so
uh I'm not sure if you're familiar with
him but Andrew yang is the presidential
candidate currently running yeah with
the math in all capital letters and his
hats as a slogan Isis stands for make
America think hard okay I'll vote for it
so and his name rhymes with yours yang
strang so but he also loves math and
then he comes from that world but he
also looking at it makes me realize that
math science and engineering are not
really part of our politics right
political discourse about political a
government in general yeah what do you
think
that is well what are your thoughts on
that in general well certainly somewhere
in this system we need people who are
comfortable with numbers comfortable
with quantities you know if you if you
say this leads to that they see it it's
undeniable but isn't it strange to you
that we have almost no I mean I'm pretty
sure we have no elected officials in
Congress or obviously the president yeah
that is either it has an engineering
degree or a mess yeah well that's too
bad you know a few could a few who could
make the connection yeah it would have
to be people who are at the door who
understand engineering or science and at
the same time can make speeches and and
lead yeah inspire people yeah yeah you
were speaking of inspiration the
president of the Society for industrial
applied mathematics oh yeah as a major
organization in math and Padma what do
you see as a role of that society you
know in our public discourse right yeah
so well it was fun to be president at
the at the time of years year two years
around around 2000 his hope as president
of a pretty small society but
nevertheless it was a time when math was
getting some more attention in
Washington but yeah I got to give a
little 10 minutes to into committee of
the House of Representatives
talking about why mint and then actually
it was fun because one of the members of
the house he had been a student had been
in my class what do you think of that
yeah as you say a pretty rare most most
members of the House have had a
different training different background
but there was one from New Hampshire who
who was my friend really bye-bye being
in the class
yeah so that those years were good then
of course other things take take over
and importance in Washington and maths
math just at this point is not so
visible but for a little moment it was
there's some excitement some concern
about artificial intelligence in
Washington now yes about the future yeah
and I think at the core of that is math
well it is yeah yeah but maybe it's
hidden maybe he's wearing a different
hat but uh well artificial intelligence
and and particularly can I use the words
deep learning it's a deep learning is a
particular approach to understanding
data again you've got a big a whole lot
of data where data is just swapping the
computers of the world and and - and
understand it out of all those numbers
to find what's important you know in
climate in everything and artificial
intelligence is two words for for one
approach to data deep learning is a
specific approach there which uses a lot
of linear algebra so I got into it I
thought okay I've got to learn about
this so maybe from your perspective and
I asked the this most basic question
yeah how do you think of a neural
network what is it and you're on the 1
yeah ok so can I start with a idea about
deep learning what does that mean sure
what is deep learning what is deep
learning yeah so so we're trying to
learn from all this day that we're
trying to learn what's important what
was some What's it telling us so you've
you've got data you've got some inputs
for which you know the right outputs the
question is can you see the pattern
there can you figure out a way for a new
input which we haven't seen to to get
the to understand what the output will
be from that new input so we've got a
million inputs with their out
so we're trying to create some patterns
some rule that'll take those inputs
those million training inputs which we
know about to the correct million
outputs and this idea of a neural net is
part of the structure of the of our new
way to create a create a rule we're
looking for a rule that will take these
training inputs to the known outputs and
then we're going to use that rule on new
inputs that we don't know the output and
see what comes the linear algebra is a
big part of defining a finding that rule
that's right linear algebra is a big
part not all the part people were
leaning on matrices that's good still do
linear is something special
it's all about straight lines and flat
planes and and and data isn't quite like
that you know it's it's more complicated
so you got to introduce some
complication so you have to have some
function that's not a straight line and
it not only doubt it on linear nonlinear
nonlinear and it turned out that the it
was enough to use the function that's
one straight line and then a different
one halfway that's so piecewise then he
said one piece has one slope one piece
the other piece has the second slope and
so that introduced getting that
nonlinear simple non-linearity in blue
the problem open that little piece makes
it sufficiently complicated to make
things interesting because you're gonna
use that piece over and over a million
times so you so you it has a it has a
fold in the in the graph the graph two
pieces
and but when you fold something a
million times you've got you've got a
pretty complicated function that's
pretty realistic so that's the thing
about neural networks is they have a lot
of these a lot of these that's so why do
you think neural networks
by using a sort of formulating an
objective function very not a plane yeah
function holds lots of folds of the
inputs the outputs why do you think they
work to be able to find a rule that we
don't know is optimal but it just seems
to be pretty good in a lot of cases
what's your intuition is it surprising
to you as it is to many people you have
an intuition of why this works at all
well I'm beginning to have a better
intuition this idea of things that are
piecewise linear flat pieces but but
with folds between them like think of a
roof of a complicated infinitely
complicated house or something that
curved it almost curved but every piece
is flat that that's been used by
engineers that ideas been used by
engineers is used by engineers big time
something called the finite element
method if you want to if you want to
design a bridge design a building design
airplane you're using this idea of
piecewise flat as a good simple
computable approximation to pay your you
have a sense that that there's a lot of
expressive power in this kind of
piecewise linear yeah well that's
combined together you use the right word
if you measure the expressivity how how
many how complicated a thing can can
this piecewise flat guys express the
answer is very complicated yeah what do
you think are the limits of such
piecewise linear or just neural networks
its passivity of nool nose well you
would have said a while ago that they're
just computational limits it you you
know you the problem beyond a certain
size a supercomputer isn't going to do
it but that does keep getting more
powerful so that's that limit has been
moved to allow more and more complicated
surfaces so in terms of just mapping
from inputs to the outputs looking at
data yeah what do you think of you know
in a context in your networks in general
data is just tensor vectors matrices
tensors how do you think about learning
from data what how much of our world can
be expressed in this way how useful is
this process is the I guess that's
another way to asking what are the
limits of this well that's a good
question yeah so I guess the whole idea
of deep learning is that there's
something there to learn if the data is
totally random just produced by random
number generators then the we're not
going to find a useful rule because
there isn't one so the extreme of having
a rule is like knowing Newton's law you
know if you hit a hit a ball and moves
so that's where you had laws of physics
Newton and and Einstein and other great
great people have have found those laws
and laws of the the the distribution of
oil in a underground thing I mean that
so so engineers petroleum engineers and
understand how how oil will sit in a in
an underground basin so there were rules
now now the the new idea of artificial
intelligence is learn the rules instead
of instead of figuring out the rules by
with help from Newton or Einstein the
computer is looking for the rules so
that's another step but if there are no
rules at all for that the computer could
find if it's totally random data well
you've got nothing you've got no science
to discover it's the automated search
for the underlying rules yeah
search for the rules yeah
exactly yeah there will be a lot of
random parts a lot I'm not knocking
random because the that's there the the
the there's a lot of randomness built in
but there's got to be some basic it's
almost always signature right in most
there's got to be some signal yeah if
it's all noise then there's there's
you're not gonna get it anywhere
well this world around us does seem to
be this seem to always have a signal
some kind yeah yeah they discovered
right that's it so
what excites you more the we just talked
about a little bit of application what
excites me more theory or the
application of mathematics well for
myself I'm probably a theory person I'm
not I'm speaking here pretty freely
about applications but I'm not a person
who really I'm not a physicist or a
chemist or a neuroscientist so for
myself I like this structure and this
flat subspaces and and and the relation
of matrices columns to rows that's my
part in the spectrum so the really
science is a big spectrum of people from
asking practical practical questions and
answering them using some math than some
math guys like myself or in the middle
of it and then the geniuses of math and
physics and chemistry and who are
finding fundamental rules and doing
doing really understanding nature that's
at its lowest simplest level maybe just
a quick in broad strokes from your
perspective what is uh where does the
linear algebra sit as a subfield of
mathematics what what are the various
subfields a year okay a you think about
in relation to linear algebra so the big
fields of math or algebra as a whole and
problems like calculus and differential
equations so that's a second quite
different field then maybe geometry
deserves to be under sort of as a
different field to understand the
geometry of high dimensional surfaces
so I think am I allowed to say this here
I think this is where personal view
comes and I think math for thinking
about undergraduate math what millions
of students study I think we overdo the
calculus at at the cost of the algebra
at the cost of linear dog titled
calculus versus linear that's right and
and you say that linear algebra wins so
you can you want can you dig into that a
little bit
why does linear algebra win right well
ok I'm the viewer is gonna think this
guy is biased not true I'm just telling
the truth as it is yeah so I feel linear
algebra is just a nice part of math that
people can get the idea of they can
understand something it's a little bit
abstract because once you get to ten or
a hundred dimensions and very very very
useful that's what's happened in in my
lifetime is the the the importance of
data which does come in matrix form so
it's really set up for algebra it's not
set up for a differential equation and
now let me fairly add probability
they're ideas of probability and
statistics have become very very
important i've also jumped forward so
and that's not that's different from
linear algebra quite different so now we
really have three major areas to me
calculus linear algebra of matrices and
probability statistics and they all
deserve a important place and and
calculus has traditionally had a had a
lion's share of the time and
disproportionate share
yes but thank you proportionate that's a
good work
of the the love and attention from the
excited young minds yeah
I know it's hard to pick favorites but
what is your favorite matrix what's my
favorite matrix okay so my favorite
matrix is square I admit it's a square
bunch of numbers and it has twos running
down the main diagonal and on the next
diagonal so think of top left to bottom
right twos down down the middle of the
matrix and minus ones just above those
twos and minus ones just below those
twos and otherwise all zeros so mostly
zeros just three nonzero diagonals
coming down what is interesting about it
well all the different ways it comes up
you see it in engineering you see it as
analogous in calculus to second
derivative so calculus learns about
taking the derivative the figuring out
how much how fast something's changing
but second derivative now that's also
important that's how fast the change is
changing how fast the graph is bending
how fast it's it's curving and my
Feinstein showed that that's fundamental
to understand space so second
derivatives should have a bigger place
in calculus second mice matrices which
are like the linear algebra version of
second derivatives are neat in in linear
algebra yeah
just everything comes out right with
those guys beautiful what did you learn
about the process of learning by having
taught so many students math over the
years
whoo that is hard I'll have to admit
here that I'm not I'm not really a good
teacher because I don't get into the
exam part the exams the part of my life
that I don't like and grading them and
giving the students a or B or whatever I
do it
because it's I'm supposed to do it but
but I tell the class at the beginning I
don't know if they believe me probably
they don't I tell the class I'm here to
teach you I'm here to teach you math and
not to grade you and but they're
thinking okay this guy it's gonna you
know when's he gonna kids he's gonna
give me an A - is he gonna give me a
b-plus what what would have you learn
about the process of learning of
learning yeah well maybe to be elated to
give you a legitimate answer about
learning I should have paid more
attention to the assessment the
evaluation part at the end but I like
the teaching part at the start that's
the sexy part to tell somebody for the
first time about a matrix Wow but is
there are there moments so you are
teaching a concept are there moments of
learning that you just see in the
students eyes you don't need to look at
the grades yeah you see in their eyes
that that you hooked them that you know
that you connect with them in a way
where you know what they fall in love
with this yes beautiful world amazing
see that it's got some beauty it's yeah
or conversely yeah that they give up at
that point is the opposite the darkest a
the math I'm just not good at math and
alcohol yeah yeah maybe because of the
approach in the past they were
discouraged but don't be discouraged
it's it's too good to miss yeah I well
if I'm teaching a big class do I know
when I think maybe I do sort of I
mentioned at the very start the four
fundamental subspaces and the structure
of the fundamental theorem of linear
algebra the fundamental theorem of
linear algebra that is the relation of
those four subspaces those four spaces
yeah so I think that I feel that the
class gets it when they want to see the
what advice do you have to a student
just starting their journey mathematics
today how do they get started
Oh No yeah that's hard well I hope you
have a teacher professor who is still
enjoying what he's doing and what he's
teaching they're still looking for new
ways to teach and and to understand math
because that's the pleasure to the
moment when you see oh yeah that works
so it's s about the material you yeah
you study it's more about the source of
the teacher being full of passion yeah
more about the fun yeah there's a moment
of of getting it but in terms of topics
linear algebra well that's not my topic
but oh there's beautiful things in
geometry to understand what's wonderful
is that in the end there there's a
pattern there there's their rules that
that that are followed in biology as
there are in every field you describe
imitation as as a hundred percent
wonderful except for the great stuff
yeah and the grades were great yeah when
you look back at your life yeah what
memories bring you the most joy and
pride well that's a good question I
certainly feel good when I maybe I'm
giving a class in in 1806 that's mi
t--'s linear algebra course that I
started so sort of there's a good
feeling that okay I started this course
a lot of students take it quite a few
like it yeah so I'm I'm sort of happy
when I feel I'm helping helping make a
connection between ideas and students
between theory and the reader yeah it's
I get a lot of very nice messages from
people who've watched the videos and
it's inspiring I just not maybe it's
take this chance to say thank you
well there's millions of students who
you've taught and I am grateful to be
one of them so good birth thank you so
much has been an honor thank you for
talking to it was a pleasure thanks
thank you for listening to this
conversation with Gilbert Strang and
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some closing words of advice from the
great Richard Fineman study hard would
interest you the most in the most
undisciplined irreverent an original
manner possible thank you for listening
and hope to see you next time
you