/sci/ - Science & Math

>>11831690
>Is it some kind of array?
Yes. Essentially an n-d array.

>>11831690
matrix is a nice gimmick for rank-2 tensors, but it is just a comfy blanket, it all falls apart when the rank goes higher than 2.
this guy's style is silly, but bear with it, it gives a good general picture of tensors for beginners
https://youtu.be/bpG3gqDM80w

Tensor is as much of an array as a triangle is an array of numbers. You can even consider proofs to be arrays of numbers. The fact that someone can be reduced to an array of numbers in some particular situation with some particular choices, doesn't mean it's "essentially just an array of numbers". Only low -IQ mathlet with no conception of higher order concepts can think that way.

>>11831718
>Only low -IQ mathlet with no conception of higher order

It's quite contrary. People with higher iq are better at generalizing aren't they? You can't call someone low-iq for him making higher generalzation. In short, you are contradicting yourself.

>>11831757
Based. Fuck autistic CSfags who are too stupid to properly understand anything.
An [math] (r,s) [/math] tensor is a multilinear map [math] (V^*)^r\times V^s \to \mathbb{F} [/math] where [math] V [/math] is a vector space over [math] \mathbb{F}, [/math] and [math] V^* [/math] is the dual of [math] V. [/math]

>>11831690
tensor is a certain abstract quantity which can be represented by an array of numbers, in general more than 2-dimensional, once coordinates have been chosen. if we change the coordinates, the numbers also change, and they do so according to a certain transformation law. physicists and engineers usually take this transformation law as the definition of tensors which leads to the phrase "tensor is something which transforms as a tensor".

https://www.youtube.com/watch?v=uaQeXi4E7gA

>>11831757
BASED BACHELOR DEGREE

and he's right, you know

>>11831690
A tensor is an element of a tensor product, being a space of multilinear functions.
/thread

>>11832043
>Writing an incorrect definition, then threading yourself, even though a better, correct definition is already written.
Retard.

>>11831690
OP is a CS faggot.

>>11832060
>>>11832043 (You) #
>>Writing an incorrect definition, then threading yourself, even though a better, correct definition is already written.
>Retard.
Brainlet. My definition is more correct.

>>11832115
>An element of a tensor product
A tensor product is not a set, and you haven't defined a tensor product.
>being a space of multilinear functions
You don't specify what you are referring to here.
Also, you don't make reference to rank.
Your definition is suggestive of a high schooler who read an article about tensors a month ago and now thinks he's the authority on them.

>>11832144
stop this madness. the anon's explanation is awful, but nothing he wrote is incorrect.

>>11832159
Fine, I'll change my contention to that.

>>11831690
In general, a tensor is map from vectors to vectors. Depending on the number of indices, it might be a map from vectors to numbers or from vectors to another tensor.

Tensors eat vectors.

Also, if you want a deeper understanding of tensors, then look at the 3D stress tensor. It has a nice wiki that explains how each component of the tensor is the pressure or shear force on each face of a cube pushing in each direction. Then you can contract it with a vector and see what happens, and then it's pretty easy to understand what it means when I say "tensors eat vectors."

a (p,q) tensor can be thought as some abstract thing eats q vectors and returns p of them, or eats p+q and gives you a number, or any arbitrary combination of which arguments to eat or spit out vectors such that its always linear in each of the p+q arguments

the rigorous concepts of multilinearity, dual space and isomorphisms are what let you do this and let you represent them as a p+q dimensional array of numbers

>>11832200
I should also say that "returning 0 vectors" means returns a number/scalar

i.e. a billinear form/metric is (0,2) because it eats 2 vectors and returns their inner product which is a number = 0 vectors returned, but with extra structure you can shuffle things around to a (1,1) endomorphism/linear map which you supply 1 vector and are given 1 vector, both are represented by 1+1=2+0 dimensional arrays.

>>11831690
>is it some kind of array
It would be recursively defined using arrays, but they are not themselves exactly arrays or matrices. They are a certain generalization of vector spaces whose construction will become obvious with study into linear algebra despite remaining annoying to work with
>>11831757
>tee here implementation is the same as the definition
I bet you think CS majors would say the approximation for inverse square root is the definition of an inverse square root.

>>11832236
>>tee here implementation is the same as the definition
>I bet you think CS majors would say the approximation for inverse square root is the definition of an inverse square root.
Dude just look at
>>11831692
clearly a lot of people think that tensors are just multidimensional arrays of numbers

Why do you even have to ask?
It's obvious just from your picture that a tensor is an item with N dimensions where each dimension can contain a defined amount of values

>>11832308
That's totally wrong and retarded.

>C#
You have bigger problems in your life to solve buddy

>>11832308
nope

>>11831794
I'm retarded but curious, do you mind elaborating?

>>11832322
https://en.wikipedia.org/wiki/Multilinear_map

>>11832322
I'll assume you already know what a vector space is.
If [math] V [/math] is a vector space over a field [math] \mathbb{F}, [/math] then its dual space [math] V^* [/math] is the space of linear maps (called functionals) that take elements of [math] V [/math] to elements of [math] \mathbb{F}. [/math] A concrete example would be to think of elements of [math] V [/math] as being column vectors, and then elements of [math] V^* [/math] are row vectors - check that under usual matrix multiplication, a row vector premultiplying a column vector results in a scalar.

Now a map [math] \phi:V_1\times V_2\times ... \times V_n\to\mathbb{F} [/math] is multilinear if it is linear in every variable - e.g. recall that [math] \phi(v_1,v_2) [/math] is linear in its first entry if [math] \phi(av_1+w,v_2)=a\phi(v_1,v_2)+\phi(w,v_2), [/math] where [math] a\in\mathbb{F}, [/math] and [math] w,v_1,v_2\in V. [/math]

So now suppose you have some [math] (r,s) [/math] tuple, [math] (w_1^*,w_2^*,...,w_r^*,w_1,w_2,...,w_s), [/math] where the [math] w_i^* \in V^*, [/math] and the [math] w_j\in V [/math] (note that e.g. [math] w_1 [/math] isn't necessarily related to [math] w_1^* [/math] ). Then an [math] (r,s) [/math] tensor would act on that tuple and return a scalar, i.e. an element of the field [math] \mathbb{F}. [/math]

>>11832322
>>11832398
Now given some vectors [math] v_1,...,v_r\in V [/math] and [math] v_1^*,...,v_s^*\in V^*, [/math] we can define an [math] (r,s) [/math] tensor as [math] v_1\otimes...\otimes v_r\otimes v_1^*\otimes ...\otimes v_s^*, [/math] which acts on an [math] (r,s) [/math] tuple (in post above) as [eqn] v_1\otimes...\otimes v_r\otimes v_1^*\otimes ...\otimes v_s^*(w_1^*,...,w_r^*,w_1,...,w_s)=w_1^*(v_1)...w_r^*(v_r)v_1^*(w_1)...v^*(s)(w_s), [/eqn] recalling that functionals act on vectors to give a scalar. Now an [math] (r,s) [/math] tensor needn't act only on an [math] (r,s) [/math] tuple. It can act on any tensor - for example [eqn] v_1\otimes...\otimes v_r\otimes v_1^*\otimes ...\otimes v_s^*(w_1,...,w_s)=v_1^*(w_1)...v_s^*(w_s)v_1\otimes ...\otimes v_r. [/eqn] There we see an [math] (r,s) [/math] tensor acting on a [math] (0,s) [/math] tuple (or a [math] (s,0) [/math] tensor - see below) to give us a [math] (r,0) [/math] tensor. Hopefully you can see what rank you get when it acts on an arbitrary tensor.

Now just note that the set of tuples [math] (w_1^*,...,w_r^*,w_1,...,w_s) [/math] has a one to one correspondence with the set of [math] (s,r) [/math] tensors, [math] w_1\otimes ...\otimes w_s\otimes w_1^*\otimes ...\otimes w_r^*, [/math] which shows why it is natural to allow tensors to act on arbitrary tensors.

>>11831692
fpbp
>>11832398
>>11832276
>>11832236
>>11832313
>>11832320
>>11832175
>>11832043
>>11832060
>>11832115
>>11831757
>>11831794
>>11831718
>>11831702
>define tensor in C#
>in C#
>b-but it's not an array!
Wanna know why I know for a fact you're the same fags who refuse to use serious programming languages even if it will get your work done faster?

>>11832398
>>11832455

Is this what mathfags find interesting?

This is just pure autism

>>11832470
>Is this what mathfags find interesting?
No
>This is just pure autism
Yeah, no shit; it's basically a single definition padded with elaboration as anon asked for.
The definition right here >>11831794 demonstrates about the amount of thought that a mathfag would put into it.

>>11832469
how to BTFO /sci/

>>11832398
>>11832455
I've done a lot of linear algebra between QM, but the notation is just so dense for me you know? I'll try to digest this.

>>11831690
gradient of a field is a vector field. gradient of a vector field is a tensor. subsequent gradients of those vector fields are tensors of increasing order.

>>11832175
Thanks, my understanding isn't beyond an array of numbers, but this definition helps me most

>>11832469
>multiple answers saying “it’s not an array but any implementation would use them in the definition”
>hurr you guys don’t know prongrammin
You’re illiterate, stupid, or both.

>>11832276
>clearly a lot of people think that tensors are just multidimensional arrays of numbers
I do not think that they are *just* that, but if you want to break it down to the simplest description "an n-d array" is exactly what you want.

>>11832398
You fucking incel

>>11832455
why does advanced maths have so many white power symbols

>>11831690
It's a list of numbers used in a data science model. Say you have some sensors for picking up weather data and you want to find out if it's going to rain. The numbers inputted into the function from those sensors, the lists of other numbers they are used in operations with (like matrix multiplication for example), and the single number the function outputs (a list can be just 1) are all tensors.

>>11831794
/thread

Some intuition:

Break down into covariant and contravariant cases.

1d) A map V->F is basically an element of V*
1d) A map V*->F is basically a vector
2d) A map VxV->F is a bilinear map, can be thought of as an abstraction of area in some sense

For other cases, there's lots of ways to think about it by "(un)currying". For example, a map VxV*-->F can be thought of as V-->Hom(V*,F)=V**=V, so is basically an endomorphism.

They're literally just covariant and contravariant multilinear maps involving a single vector space, its dual, and underlying field.

>>11834002
>It's a list of numbers used in a data science model.
Lol. Are you retarded?

> The numbers inputted into the function from those sensors
Which weather sensor has a digital data input? WTF are you on about?

> the lists
Lol.
A list is not a tensor you fucking idiot. They are not even arrays.

>>11831690
A tensor is an object that transforms like a tensor.
t. physichad

>>11834002
Wtf

>>11831690
It's a generalization of matrices.

Can every tensor be expressed as an n-dimensional array? Yes or no? I don't care about whether that's the actual definition. I just want to know if it's equivalent.

>>11831794
So how do you do this in code?

>>11834984
A tensor can be described through its actions upon the basis vectors, so you can just write down the definition. For the finite dimensional case the dual is just the Vector space of linear maps from a given vector space to the reals. So you have a mapping which goes from a tuple of elements of the duals and a tuple of elements of the vector space to some arithmetic type, this is really simple stuff.

>>11831690
>C#.
>>>/g/et out and stay out

>>11831718
what is a tensor but not an array? let’s start with rank 2 / matrix to keep it simple

>>11831794
>understanding
>while it’s purely knowledge based

>>11834984
I'd assume it's pretty straightforward. I don't know much about coding, and I only really know C, but I think you can create a struct, with two sets of arrays - one for the covariant, one for the contravariant indices. It's obvious how you can then define multiplication, etc.
Coding with tensors is not really interesting at all; it's just menial busy-work.

>>11835014
the linear map [math]\frac{d}{dx} \colon C^{\infty}(\mathbb{R}) \to C^{\infty}(\mathbb{R})[/math]

>>11835040
what is this?
and why does it not look like anon’s definition of V* x V -> F

>>11835047
tensor-hom adjunction. any map [math]V^* \times V \to \mathbb{F}[/math] is the same as (naturally corresponds to) a map [math]V \to V[/math].

>>11835051
that is neat. But wouldn’t this lose some form of information from F?

And your example is basically an array that contains operators instead of numbers right?

>>11835054
>But wouldn’t this lose some form of information from F?
no, the underlying field is an information which comes with the vector space V
>And your example is basically an array that contains operators instead of numbers right?
no. tensors are not "basically an array" and this is no exception. this anon (I'm not him) >>11831718 is right and his analogy with the triangle is spot on.

>>11835056
not that guy, but how is the triangle example spot on? *every* triangle can be reduced to a set of numbers, not just some particular cases. whether this is a useful interpretation is not directly important

>>11835056
well if I understand your example correctly, the only thing that prevents me from writing it as an array is that it contains infinite elements. which really is “basically an array”

>>11835066
btfo

>>11835064
>*every* triangle can be reduced to a set of numbers, not just some particular cases.
in the exact same way as *every* tensor can be reduced to a set of numbers
>>11835066
is it so incomprehensible that just because you can represent something as an array doesn't mean it's "basically an array" ?

>>11835064
>>11835066
>>11835014
Let me write down
(2,3,6,1,4123,512313,621323,3213123,6422,12323,232). Tell me what mathematical object this represents.

>>11835096
>Tell me what mathematical object this represents.
A tuple, up to embedding.

>>11835096
Let me tell you what it represents: it's 2 vectors in R^5. Did you get it right?

>>11835104
Wrong! You must be an idiot!
>>11835106
Here I actually lied! It's not 2 vectors in R^5, it's actually the polynomial 2 + 3x + 6x^2 + 4123x^3 +... ! I bet you feel stupid now, don't you?

>>11835082
>*every* tensor can be reduced to a set of numbers
Which set of numbers is >>11835040?

>>11835066
If you were to write down that set is 5 in it?

>>11835107
>it's actually the polynomial 2 + 3x + 6x^2 + 4123x^3 +... !
The tuples can be embedded into the polynomials, so clearly I was right.

By the way, this is how polynomials are usually implemented in software, as tuples.

>>11835107
I actually lied here as well! It's neither a polynomial nor a pair of vectors in R^5. It's actually a 2-simplex (a triangle) embedded in R^3 that is the convex hull of the points with coordinates (6,1,4123), (512313,621323,3213123), (6422,12323,232)! How crazy is that! You must be a retard if you couldn't immediately spot that this is a triangle.

>>11835110
Literally everything in maths can be represented as an array of numbers you fucking autistic retard, that's my whole fucking point. The fact that something can be embedded as an array of numbers doesn't mean that it's essentially an array of numbers. You lose almost all information the moment you encode it as an array of numbers all of which you need to keep in your head to be able to understand what is going on. If that were not the case you would be immediately able to say what mathematical object I mean by
(2,3,1,2,3,4,122,5124,2323)
Holy fucking shit the absolute state of this board.

>>11835111
Still a tuple up to embedding.

>>11833087
>Not an array
Hold on let me just

>>11835109
pick some basis [math]\alpha = (f_i)[/math] of [math]C^{\infty}(\mathbb{R})[/math], put [math]a^i_j = i[/math]-th coordinate of [math]\frac{d}{dx}f_j[/math] with respect to [math]\alpha[/math]. [math](a^i_j)[/math] is the array.

>>11835115
>Literally everything in maths can be represented as an array of numbers
See >>11835109.

>>11835122
>See >>11835109.
see >>11835121

>>11835118
;)

>>11835121
But you can not reconstruct the tensor from the array alone. Clearly they can not be identified.
Try again.

>>11835124
>>>11835122 (You) #
>>See >>11835109 (You) #.
>see >>11835121 #
See >>11835129

>>11835118
(;

>>11835096
depends on what the brackets entail, for me it’s a standard vector.

>>11835134
For me, it's a N=11, tuple.

>>11835122
You're missing the fucking point. Whenever you have a well-defined mathematical object you can encode the definition to numbers. I can encode the definition "let x be the smallest natural number such that x has more than 300 distinct divisors". I can translate every character into a number and that way faithfully represent this number as an array of numbers, each of whose entry represents a character of my definition.
The point that I'm making is that while this is used to implement computations with a lot of mathematical objects, the fact that you can do it in no way implies that the mathematical object itself is merely the numerical representation that your implementation uses. Implicit in this implementation is the very numerous assumptions that you hold in your head of what the numers in your implementations actually represent. The numbers just by themselves are meaningless. As I've demonstrated in the previous example >>11835096
>>11835106
>>11835107
>>11835111
, you can write down an array of numbers that can represent a vast multitude of completely different mathematical objects. Just by looking at the array of numbers it is IMPOSSIBLE to tell what mathematical object it is. This means that mathematical objects such as vectors, triangles, tensors are NOT just arrays of numbers. They can be represented and implemented BY arrays of numbers just like literally everything else in maths but that IN NO WAY implies that arrays of numbers is all that they are. The most important part of the object is not the array of numbers but the implicit meaning you hold in your head of what those numbers represent.

>>11835129
I don't see what's your point now. if you got confused, my point is that every tensor can be represented by an array, but it's not an array per se.

>>11835135
what’s the difference?

>>11835138
The implementation.

>>11835115
so every tensor is an array but not every array is a tensor?

>>11835137
>I don't see what's your point now.
What you wrote down is a tuple, up to embedding.
Which means it does not carry the "meta" information about it.

>>11835136
and?

>>11835140
>so every tensor is an array
A tensor is not an array. A given tensor can be REPRESENTED by an array in infinitely many ways, just like literally every other well-defined mathematical object. But the moment you represent it as an array, a stranger looking at the array would have no idea what it is unless you tell him that it's actually a REPRESENTATION of a tensor.

>>11832398
>>11832455
this is why I hate math. All the definitions are fucking autistic shit just throwing symbols and big words around which give you no idea of how it works. Then you do one example and find out you just do some stupid simple shit and it all makes sense and you wonder why the fuck mathfags have to make it sound so complicated.

>>11835142
>What you wrote down is a tuple, up to embedding.
>Which means it does not carry the "meta" information about it.
in other words, it can be represented as an array, but it's not an array per se. that's what I've been saying the whole time.

>>11835145
You have yet to define a difference other than the name, doesn't sound very scientific to me.

>>11835147
Why did you say I was wrong and called me an idiot? In >>11835107.

>boss asks you to code an infinite dimensional vector

>>11835152
not me.

>>11835149
An array can not be uniquely identified with a tensor. Neither can a tensor be uniquely identified with an array.

>doesn't sound very scientific to me.
Mathematics is not a science.

>>11834984
Like this
>int myArray[10][10][10]

>>11835145
so i tell him it’s a tensor then it is the same as an array?
btw isn’t this distinction literally the same as >>11835051?

I really only wanted to know if I can represent every tensor as array in principle or if there are exceptions.

>>11835153
let x = (1, 2, ..)

This is valid Haskell by the way, there is no issue with representing infinite vectors.

>>11835157
>They are not equal
So since it's not differentiable in code, show me in memory how it is different

>>11835146
mostly to make the stuff as widely applicable as possible. shit’s defined in a way to be able to work with any abstract objects as long as they fulfill the necessary requirements.

>>11835166
>boss asks you to take a fourier transform of that

>>11835169
>>They are not equal
Where did I say that?

>>11835169
>So since it's not differentiable in code
A linear mapping is of type list of arithmetic type -> list of arithmetic type an array is of type list of list of arthimetic type.

>show me in memory how it is different
Look at the type signatures? They will be stored with the data.

>>11835173
Ask him for a better computer since it takes so long.
Obviously you can write it down, it will just take an infinite amount of time...

>>11835162
>btw isn’t this distinction literally the same as >>11835051 (You)?
no, because the sets {maps V -> V} and {maps V* x V -> F} are isomorphic so-called naturally. the correspondence is cannonical, roughly in the sense that it doesn't require making any arbitrary choice along the way. on the other hand, representing a tensor as an array requires choice of a basis.
in other words, if you ask two people to give you an array for a tensor, you will get two different answers and there's no reason to prefer one or the other. if you ask them to transform a map V->V into a map V*xV -> F you get the same answer.

>>11835162
>so i tell him it’s a tensor then it is the same as an array?
if you tell him it's a tensor, the type of the tensor, the vector space and the basis with respect to which is the tensor represented, ONLY then he can reconstruct the tensor.

>>11835162
>really only wanted to know if I can represent every tensor as array in principle
You can literally represent every well-defined object as an array. You can represent this whole thread as just an array of numbers.

>>11835197
>You can represent this whole thread as just an array of numbers.
And in fact that is exactly what your computer is doing.

>>11835200
Exactly.

>>11835197
Yes, but is that an intuitive form of storage? The question is really more like "Is an array an intuitive way to represent tensors?".

>>11835192
What do you mean by type? The dimension and size? That should be implied in the array already. And the basis should also have an intuitive (obvious) representation as an array. I really don't see why you guys have a problem with this.

>>11835248
>What do you mean by type?
Have you not done any programming at all? How can you not know what a type is? At least on an intuitive basis everybody who has done the tiniest bit of programming should know about it.

>The dimension and size?
That is a small part of it, but not the most essential quality.

>And the basis should also have an intuitive (obvious) representation as an array.
Please, tell me an obvious representation of a basis for a Lebesgue space...

>I really don't see why you guys have a problem with this.
You seem to not understand extremely basic concenpts like a "type", of course then you can not understand the issue.

>>11835248
>type
google what is (p,q)-tensor
>And the basis should also have an intuitive (obvious) representation as an array
almost never does a space come with a preferred system of coordinates

>>11835181
ok, but if I have a specific input and a specific transformation, the tensor will always be an array, right?

>>11835197
give me a topology as array

>>11835330
Which topology? I will give you the array that represents it.

>>11835325
a tensor is represented by an array if bases of domain and codomain have been chosen. if there are no spaces, bases etc. (e.g. if you're just literally coding arrays as a way of storing numbers systematically without any relation to geometry), then you're not talking about tensors. maybe CSfags call it tensors, but it's simply wrong.

>>11835330
If you can represent a space as an array, which you obviously can, then you can also represent the topology by an array.

>>11835333
I want a donut cow plz

>>11835336
I think I get it now, thanks!

>>11831757
What's the deal with the fucking dog shit kerning?

>>11835316
I obviously know what a type in programming type is, but I thought we were talking about mathematical objects and not programming?

>>11835346
What is kerning? Define it in C#. Is it some sort of array?

>>11835352
>I thought we were talking about mathematical objects and not programming?
Types encode the "meta" properties of an object. Both in a programming language and in mathematics. The programming concept originates in mathematics.

>I obviously know what a type in programming type
Then you should know what the difference between a linear map and an array is.
A linear map is a function, an array is data stored in a certain way.

>>11835336
Okay but literally any quantity, to be represented in a computer, must implicitly be represented with the assumption of its field and vector space as well as magnitudes and units, for example a 3-dimensional velocity can only be encoded in a computer with the assumption of domain and codomain. And a 3-dimensional velocity is just an array, in terms of how it’s actually encoded, so a tensor is an array

>>11835419
>must implicitly be represented with the assumption of its field and vector space
No. Of course not.
The type is always explicitly associated with the data, that is the only way a computer would know what to do with it.

>for example a 3-dimensional velocity can only be encoded in a computer with the assumption of domain and codomain
But the domain (which is part of its type) has to be *explicitly* stored.

>And a 3-dimensional velocity is just an array, in terms of how it’s actually encoded
No. They have cleary different types and types are stored, well, at least the compiler uses the types to create a correct program where each object is associated with its correct type.

>>11835419
Youre a moron.

>>11835459
So, a tensor is just a specific type of array?

>>11835419
for the hundredth time, yes, a tensor can be represented by an array. and for the hundredth time, no, that doesn't mean a tensor IS an array.

>>11835487
>So, a tensor is just a specific type of array?
What? No! How can you even read that from what I said?

They are fundamentally different. Like a function and a list in a computer programm are fundamentally different.

>>11835487
jfl

>>11835487
Specifically it is an array of arrays.

>>11831690
Tensors are gay.

LIKE YOU!!!!!!!!!!

It's literally just an array of numbers. That's it.
>but it could be an array of arrays
Those are implemented in ram as just an array of numbers.
>but I multiplied my array by a matrix, doesn't that make it special?
No, it's just a fucking array. The way you use your array doesn't change what it is.

>>11836741
Read the thread before posting, moron.

>>11832366
>multilinear algebra
do people actually do this?

>>11836741
brainlet

>>11836789
Tensors are literally the thing that transforms "multilinear algebra" into plain old linear algebra. That's the whole point.
When you have a multilinear function
f(x_1, ..., x_n) : V^n -> W for some vector space W, i.e. linear in each argument, you can uniquely characterize it as a linear function f: V tensor^n V -> W, where now the domain is a vector space instead of a set theoretic thingy.

i dunno guys, looks like an array of numbers to me!

>>11834012
wait sorry, what's a vector?

>>11838332
V is naturally isomorphic to {maps V* -> F}. At least if it's finite dimensional.

>>11836801
Can you expand that into some sort of array?

>>11835107
This gif saves the thread

>>11835152
I was mocking the people who say mathematical structures are just arrays of numbers. If that were the case, you would be able to tell from my array of numbers that the object is a certain polynomial. Obviously you couldn't, because polynomials aren't just arrays of numbers, nor are vectors and tensors.

>>11838513
>Obviously you couldn't
But my answer was still 100% correct.

>>11831757
Imagine have x y z for positions as values. Fucking shit dev make it an array of arrays.

>>11834984
With an array of numbers

>>11831690
A Tnesor is a multilinear mapping.
Sometimes (more often than not) it CAN be represented in "multidimensional matrix form" if a basis is known, finite or at least countable.
Then you can put it into a multi layered array in for example C#

>>11835160
>my
no its not. fuck off, control freak.