Just a quick question.
I have a unit vector $$e_i = [a_1,\ldots,a_i, \ldots]$$ and all the elements of $e_i$ are zero except $a_i$ and $a_i$ = 1.
What do we call such unit vectors (I mean, is there a specific terminology)? Do we call them $i^{th}$ basis vector?
Thanks
Assuming that this is a vector in either $\Bbb R^n$ or $\Bbb C^n$, the name $e_i$ is nearly universal in mathematics texts and papers. If you need words rather than a well-understood symbol, I would say, "The $i$th standard basis vector" or "the $i$th column of the identity matrix" (with the latter being hugely less common as a description). The "$i$th canonical basis vector", as proposed by @arseniiv, also seems like a decent choice. If the entries of $e_i$ are intended to be the $0$ and $1$ elements of an arbitrary field, I'm pretty sure that "standard" or "canonical" would suffice as well. I've read a fair number of mathematics papers over the years, and such a term would never surprise me, even for a vector space over $F_2$, the field of two elements, for instance.
If $0$ and $1$ are merely meant to be elements of a ring, then "vector" is the wrong word, for $n$-tuples of elements of a ring don't form a vector space (unless you've specified some way of combining an underlying field with an element of the ring to define "scalar multiplication").