Mar 21, 2020 · This is a different way of presenting these definitions than most texts, but it's equivalent to other definitions of direct sum. In anyone's book, the sum and direct sum of subspaces are always defined; and the sum of vectors is always defined; but there's no such thing as a direct sum of vectors.

Dec 29, 2018 · Now onto my question(s). From what I've seen so far of the direct sum and tensor product, it seems like they are formal constructions to solve simple problems, for the direct sum it's a construction that allows us to add together elements from different modules and for the tensor product it's a construction that allows us to multiply together elements from …

I read the definition of direct sum on wikipedia, and got the idea that a direct sum of two matrices is a block diagonal matrix. However this does not help me understand this statement in a book. In the book I am reading, the matrix $$ \begin{pmatrix} 0&0&0&1 \\ 0&0&1&0 \\ 0&1&0&0 \\ 1&0&0&0 \end{pmatrix} $$

Dec 6, 2020 · The first definition is of an external direct sum, whereas the second definition is of an internal direct sum. See here, for example. These definitions are isomorphic. The external direct sum does result in tuples. The dimension in this case sum since the tuples are the result of the Cartesian product of the basis vectors.

Jan 5, 2016 · This is a sort of addendum to Qiaochu's answer, the purpose of which is to tie up the loose end it leaves in the sense that it only considers isomorphisms compatible with the short exact sequence associated to the surjective $\det\colon \mathrm{U}(n) \to S^1$.

For categories where the coproduct is not the direct sum (for example, when dealing with not-necessarily-abelian groups), it used to be common to refer to the direct product as the "cartesian product", the "unrestricted direct product", or even the "complete direct product" or "complete direct sum" (e.g., Hungerford offers the latter as a ...

Aug 31, 2017 · 1)V and W are infinite dimensional, but since you only take the direct sum/product once the definitions are equal. 2) No, because the element (1,1,1,1,...) is not an element in the direct sum, but it is an element in the direct product.

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Feb 16, 2015 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Jun 22, 2015 · Well, the direct product can be made between arbitrary sets, and has nothing to do with algebraic properties, while the direct sum also carries over the linear structure. However, most of the time, when you take direct product of vector spaces, you assume quietly, or directly state the existence of this linear structure on the product space ...