This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory. I do understand from wikipedia that in some cases, the tensor product is an outer product, which takes two vectors, say $\textbf{u}$ and $\textbf{v}$, and outputs a matrix $\textbf{uv ...

Next we proved that the Tensor product is unique up to isomorphism (in the sense of having this property). We did not define any higher tensor product structure such as $\otimes_{i\in I}X_{i}$. functional-analysis

Jun 6, 2017 · Over $\Bbb R$ these definitions are equivalent; the equivalence class $[v\otimes w]$ corresponding to the tensor $\frac12(v\otimes w-w\otimes v)$. But over a vector space of characteristic 2, or over a ring where $2$ is not invertible, these definitions can vary.

Feb 3, 2019 · What you are looking for is a direct proof of the right exactness of the tensor product functor. This is usually proved by applying left exactness of hom functor. However, you can find a direct proof here. Your statement can proved very briefly by applying exactness properties of tensor functor as follows.

May 13, 2017 · The reason is that $\Bbb R$ is one dimensional over itself (same with $\Bbb C$) so the tensor product with itself is going to be one dimensional, so it will just be $\Bbb R$ again (or $\Bbb C$). Thus the above expression simplifies to what you had in your first expression.

This is usually treated using the universal property of the tensor product, as in the other answers, or in Dummit and Foote, or left as an exercise. In my opinion, leaving this as an exercise is mostly about not wanting to write this much detail. I have written this out because I can't find another source that does so.

The tensor product of two Noetherian modules is indeed Noetherian. Even better, ...

Mar 7, 2016 · I am learning about tensor products of modules, but there is a question which makes me very confused about it

Jan 25, 2011 · In some sense you can think of the tensor product as reducing the Cartesian product of a space to a new space that has a direct relation to the old. This is useful in that the new space may be isomorphic to a simpler space and/or features of the new space can be used to explore features of the old space.

Sep 28, 2012 · The Kronecker product is a particular universal bilinear map on a pair of vector spaces, each of which consists of matrices of a specified size. The tensor product is a universal bilinear map on a pair of vector spaces (of any sort). In some abstract treatments, this last sentence alone defines the tensor product.