I'm taking an independent study course on linear algebra (linear algebra II), and one of the topics my advisor and I discussed was the basis for the trivial vector space, which was the empty set. I had asked why that was the case and still didn't quite understand the reasoning but my advisor had briefly mentioned the empty sum and the empty product. That is, \begin{equation} \sum_{x\in{\emptyset}} x = 0 \end{equation} and \begin{equation} \prod_{x\in{\emptyset}} x = 1 \end{equation} (This is how my instructor wrote it, so I apologize in advance if this is not the correct notation). To summarize the question, what is the reasoning behind this concept and how does that play into the convention of the empty set being a basis for the trivial vector space?
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I guess I should say that I do get the idea that the basis for the trivial vector space is the empty set since it is not linearly dependent, since by definition there's no such vector to pull from the empty set to satisfy the homogenous vector equation, which I believe is where my instructor had mentioned the empty sum. – Joseph Quenga Jr Feb 16 '25 at 00:44
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4This question is similar to: Empty set and empty sum. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. And see https://math.stackexchange.com/questions/1951490/when-indexing-set-is-empty-how-come-the-union-of-an-indexed-family-of-subsets-o/1952303#1952303 – Ethan Bolker Feb 16 '25 at 00:59
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May be helpful, that $\bigcup\limits_{i\in\emptyset}X_i=\emptyset$. – zkutch Feb 16 '25 at 01:01