To me, a tensor is a function that maps a collection of m vectors to a collection of n vectors, such that every output is linear in each input. It's true that if you choose a basis, then you can write down an array of numbers which identifies the function, and that changing the basis causes those numbers to change in a certain way, but that's not what a tensor "is".

Of course, the beauty of mathematics is that there are many different ways of looking at the same object, so physicists can deal with their arrays of numbers, I can play with my multilinear functions, and everyone gets the same answer when we ask the same question.

It is not totally without merit if you consider the tensor bundle as associated to the principal bundle of linear frames. However, that's hardly suitable as an introduction to the topic.

I'm also confused: do you think about a tensor as a vector from vector space? as a object that maps between vector spaces? or something completely different?

I realize there's a LOT of detail here - just trying to flesh out my 500,000,000 mile view a little more.

One weird consequence of this is that if A and B are vectors, the cross product (AxB) is NOT a vector in the tensorial sense. [Aside: Cross products are really weird since they are defined in the familiar way 3 and 7 dimensions only! (see http://math.stackexchange.com/questions/185991/is-the-vector...]

But usually when people say that a vector "is" a list of numbers, they mean that if you choose a coordinate system, then the coordinates of a vector uniquely identify it. This is exactly analogous to locations on the earth being uniquely identified by their latitude and longitude: we first have to make a (somewhat arbitrary) choice of coordinates. You wouldn't say that a point on the earth "is" a pair of numbers, though. The case of R^n is confusing, because if you choose the usual coordinate system, then the coordinates of a vector are the same list of numbers as the vector itself.

I like to think of tensors as functions that map some number of vectors to some other number of vectors. But you can add together two tensors (with the same number of inputs and outputs) or multiply a tensor by a number, so you can also think of them as vectors in some other vector space.

This is similar with tensors. It's even harder to talk about them, though, without the geometric/algebraic bits.

They also disagree about what "vector" and "matrix" mean for the same reason, but they can't really put that horse back in the barn. But "tensors" remain unfamiliar enough that this disagreement can come up whenever anyone says "tensor".

Here's a possible truce:

- Programmers should default to talking about "arrays"

- Geometers and physicists should default to talking about "vectors", "matrices", and "tensors"

- We all generally recognize that some (vectors|matrices|tensors) are arrays, and some arrays are (vectors|matrices|tensors), and it's not the worst thing ever if someone uses one to mean the other

- We all generally recognize that you can sometimes do damn cool things by using an array as a tensor or a tensor as an array, so there isn't that much benefit in shouting "NO THEY'RE TOTALLY DIFFERENT".

No mathematician will bat an eye if you say that a tensor can be represented by an nd-array---but if you equate them, then you're clearly missing something.

edit: that was actually a serious question, because i really don't understand this idea that programmers have a common notion of what a matrix is.

Finally, of course, a matrix is a special case of a tensor. Tensor is such a broader topic that it becomes more and more difficult to talk about it without its geometric underpinnings. This is what I was writing about above.

[0] Is a black-and-white image a matrix? I'd imagine a lot of people would have a bit of a tough time saying yes directly---and for good reason. But there's also a reasonable argument for pretending like it is, and even reasonable arguments for turning it into one!

[1] Numpy distinguishes between nd-arrays and matrices exactly right in that matrices are 2d-arrays only and are imbued with multiplication as composition of maps. In practice, many people I know who are Numpy users just think of matrix as a convenient way to overload *, though.

std::vector was an exceptionally poor choice of name...

is it so hard to just call it pseudotensor like it is? its confusing for people trying to research the subject when they come across all kinds of geometry and calculus stuff which is irrelevant to the usage here.

I think this is really a flimsy use of a technical term which does have a precise meaning in the context of mathematical physics to make the idea of multi-dimensional array seem "deeper" than it really is.

There are enough hard problems in data science - I don't think it needs to burdened with terms like this.