Suppose I have a set $S = \{\bar{u_1},\bar{u_2},...,\bar{u_{|S|}}\}$ where $|S|$ is the cardinality of $S$. I want to express a linear combination of all vectors in the set using a summation like so:
$$ \sum_{u_i \in S} a_i \bar{u_i} $$
Is this valid?
(This question is an extension of a pre-existing question: Notation of the summation of a set of numbers )
I would write $S = \{\bar u_i: i ∈ I\}$. And then $∑_{i ∈ I} a_i \bar u_i$. But note that this is an actual linear combination only if $\{i ∈ I: a_i ≠ 0\}$ is finite.
If you want the index set to be well-ordered, you may write $S = \{\bar u_α: α < |S|\}$ or $\{\bar u_α: α < κ\}$ and then $∑_{α < κ} a_α \bar u_α$. If $S$ is finite, you may use $n$ instead of $κ$ and $i$, $j$, or $k$ instead of $α$.
