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Anonymous Sun 08 Sep 00:07:48 2019 No.10954144 View Reply Original Report

Quoted By: >>10954154 >>10954169 >>10954183 >>10954208

What is the difference between tensors and matrices and vectors? I thought they were the same and a tensor was just a general term for them and the rank told you exactly what you had but my linear algebra professor told me that was wrong. Also are arrays just tensors since they can be rank 3 and above?

Anonymous

Anonymous Sun 08 Sep 2019 00:13:55 No.10954154 Report

Quoted By:

>>10954144
A matrix is a bunch of row vectors stacked on top of each other. Not sure what the fuck a "tensor" is though.

Anonymous

Anonymous Sun 08 Sep 2019 00:21:28 No.10954169 Report

Quoted By:

>>10954144
vectors are elements of vectorspaces which you can vector add and scalar multiply. both vectors and matrices you learned about in linear algebra are abstract vectors. tensors are abstract vectors that transform under coordinate transformations in a particular way so as to preserve their underlying geometric interpretation. all tensors are vectors but not all vectors all tensors.

some mathfag could probably tell you something more sophisticated, but that's the gist.

Anonymous

Anonymous Sun 08 Sep 2019 00:28:54 No.10954183 Report

Quoted By:

>>10954144
The term vector is used in different ways in different contexts. Yes a vector is a tensor of a particular rank. Yes a tensor with two indices can be represented in coordinates as a matrix.

But this language about tensors comes from physics and differential geometry. In linear algebra a vector is an element of a vector space, and matrices typically show up as components of linear tranformations between vector spaces. My guess is you said something that indicated you don't undetstand linear algebra to your professor.

In line

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Anonymous Sun 08 Sep 2019 00:33:20 No.10954195 Report

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A tensor is any object which obeys the tensor transformation law. A tensor is a multi-linear map from vectors to numbers, and in finite dimensions all such maps are tensors. In infinite dimensions, however, you have to check the transformation law because not all multi-linear maps are tensors in infinity dimensions.

>a tensor is an object such that if you put the correct amount of vectors into it, then it will spit out a number

Anonymous

Anonymous Sun 08 Sep 2019 00:35:57 No.10954202 Report

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The mathematical concept of vector has been abstracted till the point that a vector space can be any mathematical set that has certain operations(vector addition and multiplication by scalars), and the vectors are the elements. If you have a homogeneous linear ODE of degree n, the set of solutions forms a vector space. And the set of tensors of a given rank, form themselves a vector space and so technically they are mathematical vectors. I don't understand why physicists care to point this which is more of an algebraic property and not a motivation. The real motivation of tensors, is that they are functions that take vectors and output a number. In the case of stress, the thing is that because a force acting on a surface can be decomposed in different directions, this state of stress needs to be some function that has information about all these components. Yes you can arrange them in a vector, but you can no longer lool it as such in a classical sense as there are isn't one direction of stress. But the generality of saying something that takes vectors and outputa scalars (and is linear on every entry) makes it difficult to give a general view in the abstract sense. And because sometimes you can identify in a canonical way certain function with the objects they take, you can say essentialy that a vectors can be seen as a function (check dual and double dual space). More specifically, you can see tensors as things that takes either or both vectors and linear functionals and so you can construct tensors that are essentially the same as the original vectoe space.

Anonymous

Anonymous Sun 08 Sep 2019 00:37:16 No.10954208 Report

Quoted By:

>>10954144
Given A, B left R-modules, a tensor is just an object that lives in a abelian group A tensor B. This abelian group is defined as satisfying the following universal propriety:
Given a biadditive function f from AxB to some abelian group X, the tensor product of A and B is an abelian group with a function g such that (if h is the projection AxB to A tensor B) f = gh and g is bilinear.

Anonymous

Anonymous Sun 08 Sep 2019 00:44:44 No.10954226 Report

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Given a vector space V over a field k, consiser its dual V'. An (m,n)-tensor is a multilinear ( (m+n)-linear ) map from V^m × (V')^n to k.
Vectors are (0,1)-tensors since they are canonically isomorphic to their respective evaluation functionals on V'. Matrices are (1,1)-tensors since for a matrix A, a vector v, and a linear functional p, we can define a (1,1)-tensor T(v, p) = p(Av) and identify this with A.
If you have an inner product, you can identify V and V', so you can think of tensors as multilinear maps on V. Then matrices A correspond to the tensor T(v, w) = <v, Aw> and vectors v correspond to T(w) = <v, w>.
The rank of a tensor means something somewhat similar to the rank of a matrix I suppose, but it's a bit more abstract since you lose some concreteness.

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