What is a tensor anyway?? (from a mathematician)

Terrible video

How about an overview of how algebraic geometry, differential geometry, algebra and speculatively physics/ quantum mechanics and both s and g relativities might differ?
Only an overviews though with examples? ps: including metric spaces measure topology n stuff like that?

I know it's a year later, but this construction of tensors really helped to back-fill the hackneyed explanations given by engineers and physicists all over YT. I especially appreciated the definition of the R2*R3 basis at the end and its comparison to the matrix space basis. I'd LOVE more takes on tensors from your perspective, please!

Excellent!
Thank you!
❤

Thanks a lot Michael🤗😊

you loose me at the start by not explaining what the f a formal product is.

Hi there! I love your way of making things clear! Very clear and in outspoken English! You make each step easy to follow, as if it all comes about very naturally! You're a great teacher! So keep up the good work!

Wow something clicked for me with quotients when you defined the subspace of vectors that you essentially want to equal zero. I had never thought of doing this, but it makes a lot of sense with the fact that elements of the normal subgroup act like the identity when applied to a coset. Very cool.

Wow that transition from formal product to tensor product was a lot to process but I guess it's necessary for correctness and consistency.

it's funny that when we were told "matrix dimensions must agree to be multiplied", this construction makes it possible

kind of like when we were told negative numbers don't exist, or you can't take the square root of a negative

No matter how many times I see tensor products defined, I always need to go back to first principles of how a given algebra or space or object is generated. e.g. I would have guessed V*W={(a.b)*(x,y,z) a,b,xy,z in R} has dimension 5 assuming wrongly it was the span over R of (1,0)*(0,0,0), (0,1)*(0,0,0), (0,0)*(1,0,0), (0,0)*(0,1,0), (0,0)*(0,0,1)
But then I realized afterwards that scalar multiplication does not distribute.

Mathematics is a mathematical object that involves math.

This is a bit cryptic to me, as I couldn't work out what a "span" is (it doesn't seem to be the one I am accustomed to), nor what the "-" or "/" symbols meant on the objects in question.
I had to look them up on Wikipedia.

So this is a 6th dimensional space like 2 anti chiral 3 spaces or the color charge and anti color charge quark SU2 group. I finally get tensor algebra, its just number theory and simple operations. * is multiplication and + is addition.

This was The most incomprehensible video i have ever seen

Any time I read the term metric tensor my expl........

I much prefer the mathematician’s approach to tensors. Thank you for this explanation - it will help me communicate the concept!

Six months ago I would have been lost here, but I actually understood everything in this video!

Please, please do more of this! The proof (or justa a sketch of main ideas + homework exercises) of linear independence of the basis of the tensors and some notation basics how to write down (i.e. represent) and work with concrete tensor products of vectors and co-vectors; and also tensor products of more complicated structures. Like "matrix representation of a linear map", but for tensors! PLEASE!

This is really great stuff. Approaching tensors from a physics or engineering point of view is very confusing. As a pure mathematics concept it is much clearer. Keep up the good work.

as i watch the video i keep wondering if Prof. Penn has an intuitive internal mental image of the constructions his chalk writes and his mouth speaks....i notice he needs to continually refer to his written notes as he enters material on the chalkboard.....my sense is that there can be no internal mental intuitive image of what he is describing since he is describing how to construct 6 dimensional 'objects' as he calls them and there is no such thing as a 6 dimensional object or space and he does not possess the sensory or mental capacity to function or visualize 6 dimensions except as imaginary unicorns similar to the square root of negative one which does not exist in any reality.

He didn't say that you have to be familiar with the term SPAN prior to this lecture.
I have no idea what it is or what it means. Totally mystified.
I learned only about a few things that are not tensors. But there are an infinite amount of things that are similarly not tensors.

Why would we rather have c(v*w) = (cv)*w than c(v*w) = (cv)*(cw)

I would love to be able to understand why physicists call a specific permutation of sub- and super-scripts a "tensor product." I've always been able to understand the tensor product on the level you've explained, but I've never been able to understand how they're talked about in applications.

Chemical engineer here. I understood most of what you explained exept for the quotient part and stuff turning out to be zero..
Anyway that was quite different from the introduction of tensors i heard before and I understood half of both versions.
How can I create a structure so that both these halves would add up to 1 full? 😂

Ok, NOW I have absolutely no idea what tensor is 😂

wow excellent video!

i'd like more takes on the tensor product

so working in pure math makes you a pure mathematician?

It's really interesting that * was never defined concretely and that it was not equipped with any properties. If I've got this right, * can be any operation, v * w, and V * W is an abstract space? It's not quite clear to me why the dimension of the space V*W is infinity.

first explanation to actually make sense at all in any way shape or form. i cant wait to hopefully see more from u to how how maybe pixel location and rgb value need to be connected. i see the dimensions match already. obviously two different spaces.

obviously like top left 0,0,0 (black) and too left (255,255,255) white are as far as apart as they can be but there's something where like if if du and dw is small it feels similar but if du/dw is large feels like an edge. there's some relation between the spaces.

in images i feel like it should be continuous if same object, discontinous if separated into different objects (u against say ur wall in a selfi?)

Awesome video. Thank you

This is not helpful, it’s the 21st century explain the stuff more accessibly, moving on to the next source until selection favors the successful candidate.

Merry Vector Michel, and a Happy New Tensor!

Interesting! So are there perhaps other "identities" we could impose instead of the distributive rule and the scalar product rules to get different kinds of product than this tensor product? It reminds me of a "free" construction from category theory, like a free monoid having homomorphisms to every other monoid on the same underlying set

I would love more videos about tensors and tensor products.

I don't see how the formal product needs to have infinite dimension. You said it created a vector space, and so clearly it binds tighter than scalar multiplication, but scalar multiplication on the instances must work. 2(1,2)*(3,4,5) would equal 2((1,2)*(3,4,5)) and thus need a representation without the leading scalar. Treating it element-wise (bijective to R5 with weird notation) gets you all the vector properties (which it must have, since it's a span) and no weird side effects. Your "nice-to-have" properties don't seem like they should be a thing at all, because they fail to extend the scalar properties to the entirety of the {object with the star in it}.

so, how does span(R2) relate to R2 * R3, where the * also includes this "span" operation?

Just as when dealing with arbitrary real numbers, you can only say things about this arbitrary star (from a set of functions) if it holds for all star! Gl with that is the point lol

Maybe do a series on tensors. This just launched in to things with out definitions.

Formal product is formally pedantic and it's amazing

Nice explanation, friendly for beginners. The best way to define the tensor product, from my point of view (there are at least 4 different definitions). BTW, as a confusing example of non-equal elements from U*V, I would suggest elements 0*w, v*0, 0*0 and 0 (these vectors are different in V*W).

Thanks (from a hopefully future mathematician)

Love this man

amazing , please teach more about Tensors

A nice observation on formal products is that 0∗0 is not the zero element of V∗W, even if V and W are both zero-dimensional (this is also the only case in which V∗W has finite dimension, rather than being uncountably infinite-dimensional); also, it's trivial that if V or W is zero-dimensional, then V⊗W is zero-dimensional, so this is one way in which the equation dim(V⊗W)=dim(V)dim(W) works even if W or V (respectively) is infinite-dimensional.

Generally, dim(V∗W)=card(V)card(W), which is only truly interesting if V and W are both trivial or each one is over a finite field.

I would like more takes on the tenor product from different points of view. Like show a little about how it enters your real world, in representation theory.

As a mathematician I rather want a space K, such that all multikinear maps factor through K. It would be nice If such a space exists and is unique...