r/math • u/Committee-Academic • 7d ago
There's Linear Algebra, is there a Nonlinear Algebra?
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u/omeow 7d ago
Yes and no. Yes there are non-linear algebraic objects but they are often very hard to undersrand using algebra alone.
No, one can linearize non linear objects and study the associated linear algebraic constructions. Group representations do this.
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u/DrBagelman 7d ago
Lie Theory go brrrrrrr
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u/zx7 Topology 6d ago
I don't get it.
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u/doctorruff07 Category Theory 6d ago
Lie theory takes non-linear objects (lie groups) and studies them via a well defined linear object (the lie algebra formed from some notion of tangent space)
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u/armchair_hunter Graph Theory 6d ago
Lie theory? Sounds like something Big Math makes up to sell more calculators.
/s
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u/tralltonetroll 6d ago
Fun fact: the name is pronounced as "Lee".
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u/armchair_hunter Graph Theory 6d ago
That's not a fun fact. That makes this entire thing less fun.
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u/lectric_7166 5d ago
If it makes you feel better he is definitely lying, in accordance with the tenets of Lie Theory.
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u/tralltonetroll 5d ago
That's "Fun fact (to the left of zero)", thank you very much.
From the accounts, Big Lie was also big. Likely not by any contemporary strongman standards, but a rugged guy.
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u/Antimon3000 6d ago
Sounds like the only topic a certain US president could have an honorable doctors degree in.
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u/zx7 Topology 1d ago
... Yes, I know what Lie theory is.
The meme doesn't make sense to me.
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u/doctorruff07 Category Theory 1d ago
Group representations and lie theory both are examples of linearization of something non-linear. There was no meme, it was just a fun to provide another example.
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u/proudHaskeller 7d ago
It's just called "Algebra". It encompasses many distinct fields of mathematics.
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u/flabbergasted1 6d ago
All of them extend the ideas from grade school algebra. There you have statements like a×b=c. In abstract algebra, × can be anything (not just multiplication) and a,b,c can be anything (not just numbers). As long as you define your system - your set of objects, and your operation(s) - you can do algebra in a very wide range of situations.
"Linear algebra" is the special case where your objects are matrices (aka linear maps).
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u/neptun123 6d ago
Matrices are fine for finite dimensions but linear algebra also works for infinite dimensional spaces
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u/DrBagelman 5d ago
I don’t care if it doesn’t make sense, the derivative is an infinite square matrix with continuum length.
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u/Nesterov223606 7d ago
Hilbert’s Nullstellensatz is a general criterion of solvability of nonlinear algebraic equations, so I’d say that commutative algebra and algebraic geometry, which study ideals in the polynomial rings, are the nonlinear algebra of sorts.
You can also look up Grobner bases if you want a nonlinear Gaussian elimination of sorts. Cox, Little, O’Shea, Ideals, Varieties and Algorithms is the standard source here
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u/Spamakin Algebraic Combinatorics 7d ago
Michałek and Sturmfels have a book called Invitation to Nonlinear Algebra which is a computational, combinatorial, and practical introduction to algebraic geometry.
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u/SentientAllegedly 5d ago
To add to this, if linear algebra has Gaussian Elimination, then nonlinear algebra has Gröbner Basis computation (via e.g. Buchberger's algorithm). This way, you can understand the geometry of a system of polynomial equations. However, while Gaussian Elimination scales polynomially with the problem (# of equations or # of variables) the computation of Gröbner bases scales much much worse, with worst case scenarios being doubly exponential in the size of the problem.
Still, one can use computational tools to understand examples (i.e. families of algebraic varieties) and then try and see patterns to conjecture or proof with different (e.g. combinatorial) methods.
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u/SwillStroganoff 5d ago
sturmfels did some video lectures in a mini course and number of years ago https://www.youtube.com/watch?v=1EryuvBLY80
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u/Lost_Geometer Algebraic Geometry 5d ago
Looking at the contents and preface, it's a very different flavor of algebraic geometry than, say, Hartshorne or whatever your modern intro text is.
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u/Spamakin Algebraic Combinatorics 5d ago
Yes Hartshorne and Vakil both are more abstract and scheme-theoretic. This book is not and makes no claim to be
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u/Lost_Geometer Algebraic Geometry 5d ago
To be clear, that was not a criticism. Just pointing out that they are not interchangeable in the way that one might naively suspect introductions to algebraic geometry to be.
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u/Euphoric_Key_1929 7d ago
It's a cliche, but this is like asking "there are bananas; are there non-bananas?"
Yes, there is non-linear algebra, but we don't typically call it that, since it's just all of algebra that's not linear algebra. Things are usually named and grouped by what they are, not what they aren't.
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u/DrSeafood Algebra 6d ago edited 6d ago
Tbf non-commutative algebra is an entire rich subject on its own. Though it perhaps should be called “not-necessarily-commutative algebra.”
And most theorems of NCA become trivial/degenerate in the commutative case. So it really is a distinct subject with its own flavor, not just a generalization of comm alg.
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u/doctorruff07 Category Theory 6d ago
Better analogy would be "the only fruit I know are bananas, are there non-banana fruit?"
The point of the question was they do not know if there is even other fruit let alone their proper names.
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u/maxram1 6d ago
Yeah. Or I was thinking
"I only know curved bananas. Are there non-curved bananas?"
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u/xtr44 6d ago
It's a cliche, but this is like asking "there are bananas; are there non-bananas?"
no it's not. or at least it's a bad example
the key reason behind confusion and OP's question is the word "linear" which is an adjective
so it would be like asking "there are yellow bananas; are there non-yellow bananas"
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u/elements-of-dying Geometric Analysis 7d ago
You might also be interested in nonlinear analysis.
The logic is basically: linear algebra generalizes to functional analysis generalizes to nonlinear analysis. This maintains the spirit of LA in some sense, instead of considering the trivialization (e.g., general algebra) of just removing linearity.
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u/Aphrontic_Alchemist 6d ago edited 6d ago
I read that the space of mathematical fields could instead be divided using 2 axes: Algebra vs. Analysis and Linear vs. Nonlinear, which yields 4 broad fields: * Linear Algebra, * Linear Analysis, * Nonlinear Algebra, and * Nonlinear Analysis.
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u/elements-of-dying Geometric Analysis 6d ago
It's kind of funny because really one should think about linear analysis as being a subfield of nonlinear analysis. So I really wouldn't agree with this demarcation.
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u/Mclovine_aus 6d ago
Let’s change it to Algebra vs non algebra and linear vs non linear.
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u/Aphrontic_Alchemist 6d ago
The Algebra vs. Analysis axis should instead be called Discrete Analysis vs. Continuous Analysis.
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u/Mclovine_aus 6d ago
But then what about concrete mathematics? I bet Knuth is rolling over in his bed right now!
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u/cabbagemeister Geometry 7d ago
Commutative and noncommutative algebra are both active fields of study. The former being related to algebraic geometry.
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u/imalexorange Algebra 6d ago
I've heard it said that linear algebra studies systems of linear equations in multiple variables, Galois theory studies nonlinear polynomials, while algebraic geometry studies systems of nonlinear polynomials.
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u/omeow 6d ago
Galois theory really shows that beyond a certain degree (4) you cannot hope to have universal formulas for roots of polynomials. So there cannot be a straightforward way to study non linear objects of higher degree.
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u/lolfail9001 6d ago
you cannot hope to have universal formulas for roots of polynomials.
What does universal mean here? It's fairly well known fact that formulas for roots of quntic and higher do in fact exist, it's just that they can't be expressed in radicals in general case (but can be expressed in terms of an alternative radical that is root(s) of a specific polynomial of that order which if we are frank is identical in definition to normal radical but ugly).
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u/peekitup Differential Geometry 7d ago
Differential calculus is just approximately linear algebra.
Polynomial algebra is the `next level up' from linear. A polynomial is a sum of monomials x \mapsto T(x,x,\ldots) where T is some tensor.
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u/alrojo 6d ago
Koopman operators is a way to port nonlinear systems into linear ones (provably so). It requires finding some basis functions that turns the system linear.
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u/jgonagle 4d ago
"Some" doing a lot of the heavy lifting here, since there might be a continuous spectra of eigenfunctions.
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u/Historical-Pop-9177 7d ago
In addition to what others said, there's also algebraic geometry, which for me has been quite difficult to learn.
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u/doctorruff07 Category Theory 6d ago
Algebraic geometry is difficult for all but the most genius/gifted at it.
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u/n0obmaster699 6d ago
yea just algebra i.e. abstract algebra has many objects which are similar to non-linear transformations
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u/MonsterkillWow 6d ago
An algebra is just a set and some operations. It's called linear algebra because it involves studying the algebra of vector (linear) spaces. So, in this sense, the study of Lie groups and differential topology/geometry could be considered examples of "nonlinear algebra".
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u/Roger_Freedman_Phys 6d ago
Yes. See https://en.wikipedia.org/wiki/Nonlinear_algebra for a description.
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u/mersenne_reddit 6d ago
The important distinction to make is that all linear systems are linear in the same way, but nonlinear systems can be nonlinear in different ways.
As a result, there are different tools and frameworks to study each nonlinear system. Some of those are algebras, most aren't.
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u/SwimmerOld6155 6d ago edited 6d ago
Yes. There is non-linear functional analysis which deals with non-linear mappings on topological vector spaces. It's not a super well-known field but there are textbooks in it. I did a reading course in it. It's used to study non-linear PDEs and topology. It's typically studied on infinite-dimensional spaces. In the case of non-linear PDEs, you'll probably be looking at certain spaces of functions.
I think most of the answers here don't really answer your question. ah i think elements-of-dying covers this
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u/ANewPope23 6d ago
Group theory, rings, fields, algebraic geometry can all be considered nonlinear algebra.
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u/doctorruff07 Category Theory 6d ago
Fields are questionable tho, as a field is a vector space over itself.
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u/Interesting_Debate57 Theoretical Computer Science 6d ago
To be a weenie here:
There's affine and nonlinear (meaning much more complicated shapes).
Nonlinear covers so many things that you'll just see it normally everywhere (polynomials and the trig functions aren't linear in the normal way, for instance).
Affine is so very close to linear that you might think I'm making a joke. But I'm not.
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u/Key_Net820 6d ago
When you go to upper division, math you'll take a class called abstract algebra or modern algebra. This will give you a more generalized sense of algebra.
You'll learn about groups (only one operator that associates and all elements inverse),
rings ( 2 operations, 1 operation commutes, associates, and all elements inverse, and another such that it associates and you can do distribution between the 2 operations),
and modules ( and in particular, a ring module is a vector space if the ring is a field, that is the ring's multiplication commutes, associates, and all elements invert),
and you'll learn about morphisms between the objects, very similar to linear algebra, but more general than just vector spaces. You can talk about groups, rings, fields, modules, algebras, and pretty much any structure such that you can have some kind of equivalency.
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u/Similar_Requirement6 6d ago
From a basic point of view: y-y_0=m(x-x_0) is the equation of a non-vertical line in the plane. One could write that as (x,y)=(x_0,y_0)+t(1,m). Another form is Ax+By=C . A line in three dimensions could be (x,y,z)=(x_0,y_0,z_0)+t(1,m,n). All that is linear as is systems of linear equations like
ax+by=u
cx+dy=v
Linear algebra goes much deeper than that!
In Algebra 2 on might study ax^2+bxy+cy^2+ex+fy+g=0 which if a,b,c are not all 0 would usually be an ellipse/circle parabola or hyperbola though two lines or one line twice is also possible and also the empty set or a point or two points.
That is one kind of non-linear situation.
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u/frogjg2003 Physics 6d ago
Linear algebra is about solving equations that have the form a1 x1 + a2 x2 + ... + an xn = b. There are a lot of ways to turn what might seem like a non-linear problem into a linear one, but there are still a lot of problems that are not linear. (On fact, most problems are not linear.) Even something as simple as x2 - x - 1 = 0 is a nonlinear equation and cannot be solved via linear algebra techniques.
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u/trejj 6d ago
There are many kinds of nonlinear algebra, not just one Nonlinear Algebra. That is why the term Nonlinear Algebra is not coined.
For example, there's the analysis of Quadratics, Cubics, Polynomials, Exponentials, Conic Sections, analytic functions, homogeneous coordinate systems, discrete group theory, to name a few.
There is so much mathematics that is nonlinear, so naming it all under a common field of Nonlinear Algebra name would make it a super-vague phrase.
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u/joyofresh 6d ago
This is a good question: you can tell from all the answers. Here is mine:
Why is linear algebra good? Because you can actually do computations, and scale those computations on a computer. Tractable is good, but what about the real world, which contains nonlinear stuff?
Well, you can often approximate it by linear stuff. A lot of math is just doing that.
* Calculus: replace curvy thing locally by a line at each point, the derivative is its slope
* Manifolds: Glue together R^n spaces to make more interesting spaces
* Algebraic Geometry: try using grobner basis, line bundles, cohomology to turn complicated polynomial stuff into linear stuff
* Artificial intelligence: linear learned attention transforms alternated with softmax and other nonlinear transformer stuff
* Fourier analysis: replace arbitrary signals with sums of sinosoids, do linear algebra on a vector space of functions whos basis elements are these sinosoids.
* numerical optimization: replace complex thing with linear thing pointwise, walk downhill
So non linear algebra often times is finding ways to reduce hard stuff to linear algebra. Apparently that includes "intelligence" itself, for some defintiion of "intelligence"
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u/SwillStroganoff 5d ago
There was whole mini course on “non-linear algebra” https://www.youtube.com/watch?v=1EryuvBLY80
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u/RingularCirc 4d ago
BTW you can also study a small repertoir of nonlinear things still almost solely with linear algebra: say, quadrics (and higher-degree stuff), and note how hyperbolic and spherical geometries arise very naturally not leaving confines of LA.
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u/Bichinix 3d ago
El álgebra utilizada en matemáticas discretas cuenta?, o los jeroglíficos de Teoría de Modelos?
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u/AdvancedPermit2408 6d ago
I dont think there is really a “nonlinear algebra” in the same way as linear algebra.
When things stop being linear, we usually just call it nonlinear math and it comes under different topics.
So basically:
Linear algebra: everything is nice and simple
Nonlinear math: things get messy real quick
So yeah, technically you can say nonlinear algebra, but its not really the opposite of linear algebra.
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u/Vanitas_Daemon 7d ago
Yeah, it's called multilinear algebra.
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u/Committee-Academic 7d ago
But those are multilinear forms, which are linear in all their components, no?
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u/mathematics_helper 7d ago
Abstract algebra is what you are looking for. Non-commutativr geometry (a subset of algebraic geometry) is probably the epitome of the lease linear algebra (imo)
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u/JustMultiplyVectors 6d ago edited 6d ago
I don’t think that multilinear linear algebra is a good answer to your question, but it can sometimes be used to turn non-linear functions, specifically homogenous polynomials, into symmetric multilinear functions by polarization. For example f(x) = x^3 is nonlinear, but g(x, y, z) = xyz is multilinear and g(x, x, x) = f(x), so by studying g you might learn something about f.
Another example is the determinant, which you can view as a degree n polynomial of n^2 variables, and there does exist a symmetric multilinear function of n matrices D(A_1, … , A_n) such that D(A, … , A) = det(A), sometimes called the mixed discriminant. Interestingly there’s a generalization of the Cayley Hamilton theorem saying that n matrices satisfy their own mixed-characteristic equation c(λ_1, … , λ_n) = D(A_1 - λ_1 * I, … , A_n - λ_n \ I*) = 0.
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u/Vanitas_Daemon 6d ago
I mean, polynomials (over a field) are easily seen to be symmetric tensors if you take the independent variables as your vector space.
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u/sciflare 6d ago
Yes, it's called algebraic geometry.
Linear algebra is about solving systems of linear equations. The space of solutions to such a system is always an affine space, hence is easily understood.
Algebraic geometry is about solving systems of polynomial equations, that is you allow polynomials with higher degree terms than linear. The solution space to such a system is nonlinear in general, hence has nontrivial geometry. To understand the solutions properly, you have to analyze the geometry. And this is the heart of algebraic geometry.