![]() ![]() My source for this is Lee's Riemannian manifolds book. Once you get the first part you can start thinking about tensor products of modules and the tensor algebra as a graded algebra with each grade further categorized by type (p,q). It’s a tricky concept so give it some time and thought if you get frustrated. This is probably way more than you need, so I’d suggest going and watching those videos and then coming back if you have more questions. In simpler terms, the tensor product “kills” the need to consider linearity in multiple arguments of functions by finding a quotient space that makes everything linear. ![]() In other words, every bilinear map on the Cartesian product will factor through the Tensor product as f=g∘u where u is the natural map from U×V to U⊗V. If we take a the product of two vector spaces U×V and we then map it bilinearly using some function f to some other vector space W (it’s often easiest to just think of W=ℝ), then the tensor product U⊗V is the unique object which will map linearly under a particular function g. What’s REALLY interesting about tensors though is the top-down approach which is more categorial and comes from something called a universal property. We would call this object a tensor of rank 3. We did a tensor product and then another tensor product which combines three vectors. This essentially just means it was built using more vectors than we used in the first example. This is how you get what you’ll probably see called “higher rank” tensors. Now notice that every tensor space formed from vector spaces is itself a vector space and you can inductively continue to tensor things with it. It’s “raising the dimension” of the array that you need to use to represent it. This gives you an idea of what the tensor product is actually doing in terms of components. One way to think of it at a low level when using these bases is just as matrix multiplication of Only now the basis requires all combinations of the previous basis vectors under the tensor product. Now recall that any of those vectors can be written as a linear combination of some chosen basis vectors and you have a basis for your tensor space. Where u,v, and w are vectors and a, b are scalars. You probably know what linear means, but I’ll write out bilinearity just in case: The funny symbol there is called the tensor product and what’s important about it is that it is linear in each argument. (We often build tensor spaces using vector spaces and their duals, but it’s not explicitly necessary to use the dual.) ![]() However, I’d also recommend making sure you know linear algebra fairly well.Ī tensor can be thought of in a bottom-up construction as being built out the formal symbols v⊗w where v and w are vectors or covectors. He has a great intro series that starts with the actual tensor algebra and pretty carefully builds it up from basic linear algebra. To view LaTeX on reddit, install one of the following:
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