Proper way to deal with lists and map

Question

So, I wanted to do something that I assumed would be really simple. Given two lists, $A = \{a_1, a_2, ...\}, B = \{ ... \}$, I wanted to take a function $f(a, b)$ and generate a list with $\{ f(a_1, b_1), f(a_1, b_2), ... \}$, i.e. just the cartesian product.

Of course, it's really easy, in principal, to do this:

f[#1, #2] & @@@ Outer[List, {a1, a2, a3}, {b1, b2}]

The annoyance comes in when the elements are themselves lists. In this case, everything breaks. Ridiculous things start happening.

There is a "trivial" solution; just use dummy variables in the above statement, and replace it in with what I want later. But to me this seems fundamentally wrong somehow. Surely it should be possible to work with lists of arbitrary type; and not have to "protect" your mapping functions from lists of lists.

Any thoughts? Am I missing some fundamental strategy here?

-- Edit:

To elaborate more, I'm actually trying to make the following work:

dim = 2; (* Say *)
Table[
  SomeFunc[dim, g, op, #] & /@ 
    Subsets[Range[dim], {i}], {i, 1, dim}
]

op previously was a single matrix; but now I need it to be a list of matricies. That is so say, I want to run "SomeFunc" for each matrix in that list, and also each of the results that come from the Subset function (note, of course, that Subsets returns lists.)

Answer 1

To resolve this question:

This construction would do what you want, it seems:

opList = Array[C, {3, 2, 2}]; (* list of matrices *)
sList = Array[K, {3, 4}]; (* list of vectors *)

Outer[f, opList, sList, 1]
   {{f[{{C[1, 1, 1], C[1, 1, 2]}, {C[1, 2, 1], C[1, 2, 2]}},
       {K[1, 1], K[1, 2], K[1, 3], K[1, 4]}], 
     f[{{C[1, 1, 1], C[1, 1, 2]}, {C[1, 2, 1], C[1, 2, 2]}},
       {K[2, 1], K[2, 2], K[2, 3], K[2, 4]}], 
     f[{{C[1, 1, 1], C[1, 1, 2]}, {C[1, 2, 1], C[1, 2, 2]}},
       {K[3, 1], K[3, 2], K[3, 3], K[3, 4]}]},
    {f[{{C[2, 1, 1], C[2, 1, 2]}, {C[2, 2, 1], C[2, 2, 2]}},
       {K[1, 1], K[1, 2], K[1, 3], K[1, 4]}], 
     f[{{C[2, 1, 1], C[2, 1, 2]}, {C[2, 2, 1], C[2, 2, 2]}},
       {K[2, 1], K[2, 2], K[2, 3], K[2, 4]}], 
     f[{{C[2, 1, 1], C[2, 1, 2]}, {C[2, 2, 1], C[2, 2, 2]}},
       {K[3, 1], K[3, 2], K[3, 3], K[3, 4]}]},
    {f[{{C[3, 1, 1], C[3, 1, 2]}, {C[3, 2, 1], C[3, 2, 2]}},
       {K[1, 1], K[1, 2], K[1, 3], K[1, 4]}], 
     f[{{C[3, 1, 1], C[3, 1, 2]}, {C[3, 2, 1], C[3, 2, 2]}},
       {K[2, 1], K[2, 2], K[2, 3], K[2, 4]}], 
     f[{{C[3, 1, 1], C[3, 1, 2]}, {C[3, 2, 1], C[3, 2, 2]}},
       {K[3, 1], K[3, 2], K[3, 3], K[3, 4]}]}}

The level argument 1 in Outer[] essentially tells Outer[] to treat everything in level 1 (that is, the elements of the input lists) as atomic, instead of having Outer[] treat the first list as a rank-3 tensor and the second list as a matrix.

Answer 2

Is this what you are after ?

a = {a1, a2, a3}
b = {b1, b2, b3}

Distribute[f[a, b], List]

{f[a1, b1], f[a1, b2], f[a1, b3], f[a2, b1], f[a2, b2], f[a2, b3], f[a3, b1], f[a3, b2], f[a3, b3]}

And as suggested by Artes the equivalence of solutions can be shown by:

a = Array[c, {3, 2, 2}]; b = Array[k, {3, 4}]; 
Distribute[f[a, b], List] === Flatten[Outer[f, a, b, 1], 1]

True