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The direct sum $\oplus$ versus the cartesian product $\times$

  1. (Definition) I was wondering how their definitions are different? Are they both the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components?

    How are the following concluded from their definitions:

    A direct product for a finite index $\prod_{i=1}^n X_i$ is identical to the direct sum $\bigoplus_{i=1}^n X_i$. The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. in fact, direct sum is a proper subset of direct product.

  2. (With category) I also wonder why the following is true:

    Direct sum and direct product are dual in the sense of Category Theory: the direct sum is the coproduct, while the direct product is the product.

    If I understand correctly, in category theory, coproduct is characterized by injection and product by projection. To me, both direct sum and direct product can be characterized by both injection and projection.

  3. (With matrix)Are their connection between direct products of vector spaces and of matrices (for matrices, as far as I know, their direct product is Kronecker product)? Similarly for direct sums of vector spaces and of matrices?

    If consider matrices as representation of linear mappings between vector spaces, I don't know how they are related.

    if consider a vector as a matrix, the direct product of two vectors in the matrix sense is not the same as in vector space sense, because the dimensions of the two direct products are not the same.

All quotes are from Wikipedia.

Thanks and regards!

Community
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Tim
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    For finitely many factors, the direct sum and the direct products are equal/isomorphic: in fact, they are a "biproduct". It is only when you have infinitely many nontrivial factors that they differ. See this previous question. Though the question and answers are about abelian groups, they hold mutatis mutandis for vector spaces. – Arturo Magidin Aug 10 '11 at 20:23
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    The first question is answered in the question I've just closed this question as a duplicate of. If you feel that this leaves your other questions unresolved, consider asking them separately. For the second question, look in a category other than abelian groups (such as sets). For the third question, the Kronecker product is the tensor product with respect to a basis, which is neither a categorical product nor a categorical coproduct. – Qiaochu Yuan Aug 10 '11 at 20:25
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    You can connect direct sums of matrices with direct sums of vector spaces in the following sense: if $A$ is an $n\times m$ matrix and $B$ is a $p\times q$ matrix, then $A\oplus B$ is the block diagonal matrix that has upper left block $A$ and bottom right block $q$. Interpreting $A$ as a map $\mathbf{F}^m\to\mathbf{F}^n$ and $B$ as a map $\mathbf{F}^q\to\mathbf{F}^p$, then $A\oplus B$ is the corresponding map $\mathbf{F}^m\oplus\mathbf{F}^q\to\mathbf{F}^n\oplus\mathbb{F}^p$. However, that's a bit of a stretch. – Arturo Magidin Aug 10 '11 at 20:25

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