I took linear algebra during the COVID era and I did really well and really enjoyed it, but then again it was the COVID era. So I recently bought the Axler book on Linear Algebra, but while I am reading thought it, I notice the text references the jth coordinate, term, slot, etc. and I was just confused by what the significance of this might be? I've been looking through the book to see if there was something I possibly missed and I can't seem to find anything, nor is there really anything online about it.
Just curious if it is something that I need to emphasize while I relearn this content? or is it something analytically nuanced that can be omitted for the sake of relearning the techniques?
Thanks!
edit: I will quote a specific section for reference:
"Suppose $U_j$ is the subspace for $F^n$ of those vectors whose coordinates are all $0$, except possible the $j^{th}$ slot..."
or
" $F^n$ is the set of all lists of length n of elements of F:
$F^n = [({x_1},...,{x_n}): {x_j} \epsilon \ for \ j = 1,...,n]$ for $({x_1}, ... , {x_n})\ \epsilon\ {F^n}\ and\ j\ \epsilon\ [1,...,n],$ we say that ${x_j}$ is the $j^{th}$ coordinate of $({x_1},...,{x_n})$ "