Symbolic Indexing on Symbolic Arrays

Question

Is there a way to use symbolic indices on symbolic vectors? The most barebones example I can think of is the following:

aV = Array[a,3];
aV[[m]]

which evaluates to a warning:

{a[1], a[2], a[3]}[[m]]
Part::pkspec1: The expression m cannot be used as a part specification.

Instead, the behaviour I would like to see is that this simply evaluates to the symbolic expression

a[m]

Having this functionality for symbolic indexation would be useful for many more complicated contexts to derive symbolic expressions, and Integration into the Indexed function would also help keep expressions readable.

To add another less barebones example, here I calculate a general cross product between vectors $b_l$ and $b_m$ in $\mathbb{R}^3$:

Nb = 7 ; (*length of list of R3 vectors b_i, not important*)
bV = Array[b, {Nb, 3}];
stdbasis = IdentityMatrix[3]; (* standard basis vectors in R3*)
crossProductFormula[l_, m_] = 
 Sum[stdbasis[[i]]*LeviCivitaTensor[3][[i, j, k]]*bV[[l, j]]*bV[[m, k]], {i, 3}, {j, 3}, {k, 3}]

The function works for specific l and m inputs, but it would be nice if it could generate a readable symbolic expression.

Answer 1

Will this solve your first question?

aV[m_] := Array[a, 3][[m]]

Let us check:

aV[1]

(* a[1] *)

Sum[aV[m], {m, 1, 3}]



  (* a[1] + a[2] + a[3]  *)

??

Otherwise you could have define like this:

Subscript[aW, m_] := Array[a, 3][[m]];

Subscript[aW, 2]

(* a[2] *)

Sum[\!\(
\*SubsuperscriptBox[\(aW\), \(m\), \(2\)], \({m, 1, 3}\)\)]

    
   (* a[1]^2 + a[2]^2 + a[3]^2  *)

??

Answer 2

I found a solution to the second example in this question which involves defining the arrays in question as functions:

basis[i_] := KroneckerDelta[#, i] & /@ Range[3];
bs[i_, j_] := Indexed[b, {i, j}]

However, this is not entirely satisfactory since the function 'bs' now does not have specific dimensions, and hence we cannot retrieve a symbolic row or column like $b_{l:}$ and $b_{m:}$. Further, Mathematica would not know how to, say, take the dot product of these two arrays unless we explicitly define it as a sum of precisely indexed elements.