Advice for research mathematicians

In the 19th and 20th centuries, the relative importance of matrices and linear transformations changed places. But which is more fundamental? Here is some personal advice for prospective mathematics researchers, especially amateurs, about how to approach certain questions and investigations in linear algebra.

We consider the example of mutation matrices which appear in the Dynamics on Graphs series. And we also look at the importance case of the symmetric groups S_n and how to think about them.

Here are the Wild Egg Maths Playlists (some available only to Members!)

Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...

Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...

Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...

Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...

Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...

Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...

Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...

Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

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Here are the Insights into Mathematics Playlists:

Maths and Music: https://www.youtube.com/playlist?list...

Diffusion Symmetry: https://www.youtube.com/playlist?list...

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

Playing Go: https://www.youtube.com/playlist?list...

Year 9 Maths: https://www.youtube.com/playlist?list...

Ancient Mathematics: https://www.youtube.com/playlist?list...

Wild West Banking: https://www.youtube.com/playlist?list...

Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...

Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?

list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov

Math History: https://www.youtube.com/playlist?list...

Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...

MathFoundations: https://www.youtube.com/playlist?list...

Wild Linear Algebra: https://www.youtube.com/playlist?list...

Famous Math Problems: https://www.youtube.com/playlist?list...

Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...

Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...

Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

Differential Geometry: https://www.youtube.com/playlist?list...

Algebraic Topology: https://www.youtube.com/playlist?list...

Math Seminars: https://www.youtube.com/playlist?list...

0:01

Hello everyone i'm Norman Wildberger and i'm giving you some advice for prospective

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research mathematicians which includes graduate students and also actually undergraduates who are

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interested in learning how to investigate pure mathematics in my last video or two i've been

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talking a little bit about spread polynomials context of poly number theory enlarging number

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theory to embrace polynomials or what we call poly numbers and i'd like to augment that discussion in

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in the future but i want to do it in the direction not just of polynomials but also matrices which

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is the other great development in arithmetic so we start with arithmetic with ordinary numbers

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whatever they are there's lots of different number systems and then we can move to poly numbers or

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polynomials and that's a big step up like i'm trying to convince you but there's also this

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other direction where we move to matrices or maybe better yet Maxel theory okay so um both of these

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are really great arenas for amateur mathematicians to investigate and today i want to talk about

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an interesting tension between matrices and linear transformations so this is in the direction of

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linear algebra i want you to to reconsider a basic orientation to a really fundamental

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aspect of modern mathematics so i'm going to ask the question which of these things wins

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so matrices on the one hand are very explicit and concrete and they were of course central in

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19th century mathematics so already you're there there was a well-developed theory of matrices but

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linear transformations came later on they're more implicit and abstract and are central of course

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in modern 20th century linear algebra and we can ask the question which is more important which

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should you know more about okay so this is a special case of a more general tension

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or a more general say discussion in in modern mathematics which is between the concrete

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and the abstract and typically with such things the tide goes in one direction for a while and

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then it sort of goes in the other direction so it's safe to say that the 20th century was

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a a time of considerable abstraction but now that we're moving into the 21st century i think there's

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a reaction to that and especially with the advent of modern computation people are looking to

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ground themselves more concretely in computational reality now in general i would say that the single

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most important book in this direction is the wonderful tome concrete mathematics by

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Graham Knuth and Patishnik and it's really a fabulous book that if you haven't got it you

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should get it because you can download it from the internet and you can read it you can read it sort

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of selectively and just skim through it and if you've been watching my how to solve polynomial

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equations uh series uh you'll know that uh we lean quite heavily on this on this book

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and some insights from it to solve that problem of how you actually solve polynomial equations

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there's another really important resource in this direction in the direction of concreteness

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and explicitness which i think should be very familiar to all prospective research

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mathematicians amateur and professional like and that's the online encyclopedia of integer

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sequences abbreviated o e i s and you've watched me use it in many different occasions created

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originally by Neil Sloane and contributions now of hundreds probably of thousands of people so

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it's absolutely a fabulous resource and you should um just peruse it and and play around with it and

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then you know look for opportunities of getting it to help you okay so today i want to talk about

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this tension between linear transformations and matrices doing your transformations and the usual

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story that you're probably going to learn as an undergraduate taking linear algebra courses is the

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something like the following that the central objects in linear algebra are vector spaces

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or at least that's sort of the starting point but once we've got vector spaces

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then we can consider abstract linear maps between them or maybe maps on a vector space into itself

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and these are called linear transformations and probably most lectures would say that's the linear

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transformations which are really at the heart they're the central objects of linear algebra

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so for example you know v is the space of all n by n matrices okay that's a linear space which

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is actually quite concrete another example of a vector space more abstract that rises say an

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analysis there's all kinds of vector spaces that arise there for example the lp functions on the

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n-dimensional sphere that's a lot less concrete because to actually specify an lp function on an

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n-dimensional sphere is highly non-trivial unless it's going to be a trivial kind of function and

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then but these are just the spaces and then we consider you know operators or or linear

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transformations on these spaces and and that's the usual story in terms of modern linear algebra

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if we want to make computations what we typically do is we take these vector spaces and we we find

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explicit bases of them okay once you've found an explicit basis of a vector space

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that's a typically finite or maybe an unbounded collection of vectors if it happens to be like an

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unbounded space like the lp space and it has the property that it's it's a spanning set and it's

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independent and roughly what that that amounts to is the fact that you can then identify

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an arbitrary vector in the vector space with basically a vector an explicit vector of numbers

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okay maybe it's a finite vector or maybe it's an unbounded vector in these

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unbounded situations so that allows us to move from abstract vector spaces to vector spaces of

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of vectors that are just strings of numbers like row vectors or column vectors

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and then the abstract linear transformations that we've been talking about can then be represented

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by matrices okay so the matrices come about in linear algebra by choosing bases of vector spaces

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and then representing linear transformations in terms of how they act on these various bases

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and the pleasant aspect of that is that the the operation of composition of linear transformations

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you do one and then you follow it by another that gets transformed into a product of matrices

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and there's a very explicit way of multiplying matrices that you all know uh that then uh

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you know is brought to bear on describing the the algebra of linear transformations

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so usually the story is that that linear transformations are sort of the pure or

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primary objects and a matrix is is something that we cook up perhaps a little bit arbitrarily

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in order to study it a little bit more concretely

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but it depends very much on the choice of bases in the in the linear space that we're starting with

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however in practice what happens is that whenever you want to make some non-trivial computation

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typically you you require matrices so all our algorithms are most of our algorithms run on

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matrices okay and that's that's the computational side of things now if we look at the history

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historically the matrices actually came before the linear transformations so matrices were studied in

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the 19th century and actually earlier also before the emergence of modern linear algebra before the

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notion of a vector space was crystallized and the the concept of linear transformation became

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uh quite important in fact they probably really first rose in this very practical problem of

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solving systems of linear equations which at least when you first look at it doesn't necessarily have

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much to do with abstract vector spaces and linear transformations and in particular in this in this

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drive to solve systems of linear equations the determinants

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of matrices ended up playing a particularly significant role okay so the determinant as you

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know if you have a square matrix a determinant is a number that's associated to this matrix

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and it's actually rather a complicated process of extracting this number from a matrix and

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in the 19th century there was a very elaborate and well-developed theory of determinants and matrices

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and um the standard treatise on this subject is this a lovely book called a treatise on the theory

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of determinants by actually sir Thomas Muir and i think it was first published in 1882 although it's

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been reprinted several times since then in fact including in a rather relatively new

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printing around 2011 i think so this is well acknowledged to be a fountain of interesting

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information and i do encourage you to to try to get a hold of this book it's really a lovely book

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especially for amateur mathematicians okay it has all kinds of really great stuff in it that

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you can access directly and that will help you in investigations okay the theory of matrices just

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like the theory of polynomials or polynomials is really important in in explorations in actual real

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life explorations okay so now i'm going to give you my opinion okay you know in terms of this

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controversy which you've already guessed already so let me be explicit in my opinion matrices are

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actually more fundamental and more important than linear transformations that's not to deny

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that linear transformations aren't important i don't want to say that but i'm just saying that

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matrices are more fundamental and ultimately more important and i'm suggesting that amateur

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mathematicians should move away from abstract linear algebra towards concrete linear algebra

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okay so this has advantages it first of all opens the door to many more interesting new developments

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and it also allows us to go back to reappraise existing theories okay there's lots of things

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that have been done that perhaps can be made more concrete and richer structure can be revealed

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and of course it connects much more directly with the modern computational world

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and all these fabulous algorithms that

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computer scientists and applied mathematicians have have created in to help us work with matrices

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now so from our point of view i want to reverse the the relationship between between linear

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transformations and matrices i believe that matrices have an independent existence

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aside from linear transformations so they are not intrinsically associated to linear

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transformations in the way that people usually think in fact for me a matrix determines a

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whole range of linear transformations and of quite different kinds so we do not want to

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think of a matrix as representing something else okay we want to think of a matrix as a basic

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uh object basic object that that we study on its own and it has applications but a matrix is just

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a matrix so to give you an example of this i want to give you just a couple of examples

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one of them is from my course on um well dynamics on graphs which of course also a part of this

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explore research mathematics adventure that we're going on here about

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so it's um the topic of mutation matrices in the context of uh playing games on graphs

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okay so let me just illustrate with uh an example in case you haven't been following

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this this series if you you haven't uh you might very well like to do so here's a graph with the

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four vertices one two three four here's the adjacency matrix of that graph with respect to

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those those vertices and then there's a certain game that we can play on on functions on the

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vertices of this graph these functions are called populations and this this game is played via

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mutation matrices one for every vertex of the graph so here are actually the four mutation

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matrices that uh that go along with this graph okay they're four by four matrices now this um

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say this second one here s2 okay so this mutates on a population vector so there's a

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vector that we can create basically a function just values on the four vertices p1 p2 p3 p4

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and this mutation matrix mutates this population just by multiplying this row vector by the given

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four by four matrix okay and it turns out that you can check that the action is to send this

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vector to this vector so this is encoding what we call the mutation at s2 on a general

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population so this matrix yes it does act on the space of row vectors or population vectors

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but as i've mentioned to you relatively recently this matrix also acts

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on the left on a sort of a dual kind of vector which we might call a weight vector so here's the

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weight vector w1 w2 w3 w4 it's a column vector and now the action is a left action so we take

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this column vector multiplied on the left by the same matrix then you see that its effect is this

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okay in this case uh the the entry here w2 gets changed to its negative and then the adjacent

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vertices get changed over here what happened is only the second element got changed and it was

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sort of negated and the sum of the neighbors was added to it so there's a different a different

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kind of interpretation of the same matrix s2 on the one hand it acts on populations in a certain

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fashion but in another direction another way it acts on these weight vectors in it in a different

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way so this is the mutation game up here and this is the numbers game and the interaction of these

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things is a very rich uh thing but we don't want to just think about this matrix as being

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one of these linear transformations the matrix has an independent life and it can act either

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on this space or on this space in fact if you think about it a four by four matrix acts on

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all kinds of spaces right i mean we could take say four by six matrices okay think of the space

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the vector space of four by six matrices then a matrix like this is going to act on that space by

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uh by left multiplying so you take a four by six matrix and you left multiplied by s2 you're going

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to get another four by six matrix that's going to be a linear transformation so this s2 matrix in

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principle it can act as a linear transformation on a whole range of different spaces

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okay so i'm trying to weaken this idea that's probably been imprinted on you that you start

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with a linear transformation you create a basis and then you get a matrix the matrix

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is some kind of secondary object associated to the linear transformation okay forget that okay

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think of the matrix as being fundamental and the matrix this particular explicit concrete

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4x4 matrix has a wide variety of interpretations as linear transformations on all kinds of things

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the matrix is at the heart of the story not the linear transformations from this point

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of view okay that's my point of view let me give you another example uh in this direction

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so that's the uh the nature going sort of into group theory we're talking about the

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symmetric group also in the context of this this adventure of exploring these dynamics on graphs

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so we defined a permutation matrix to be a matrix where each row and each column has exactly

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one one and the rest of the entries are zero so when n is four here is a four by four permutation

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matrix each column has a single one in it each row has a single one in it everything else is

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zero the determinant of this matrix happens to be one it's always either one or minus one

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and here's the inverse which is also a similar kind so once we have the notion of a permutation

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matrix then we can define the symmetric group to be the set of n by n permutation matrices

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and the operation is matrix multiplication so that for example the symmetric group s3 consists of

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these six matrices these are the six permutation matrices in the three by three case and the entire

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symmetric group is just this set of permutation matrices now this is a non-standard definition

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this is not usually the way the symmetric group is introduced usually the symmetric group is

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thought of in terms of permutations of a set of three things okay i want you to separate

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yourself from that that notion okay the symmetric group is better thought of as a group of matrices

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and this group of matrices yes as we've talked about it can act on row vectors but it can also

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act on column vectors and as i've explained in this series that's more or less the distinction

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between acting on objects and acting on positions and whether you're thinking about you know acting

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on objects or whether you're thinking about acting on the underlying positions there's a

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basic symmetry there okay so when we stick with just the matrices themselves we're not we're not

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committing to one interpretation or another we have both interpretations readily at our disposal

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all right so i'm i'm hoping that this little bit of advice will just reorient you a little bit

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and get you to appreciate the importance of matrices the value of learning about

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matrix manipulations and algorithms for matrices and also to incline you to

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look whenever you're doing research about ways of connecting with matrices

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i'm going to be showing you an interesting very powerful example of that in my next advice video.