Getting keys of an 'Association'

Question

Suppose you have an association of the type:

a = {1, 0, 0, 0}; b = {0, 1, 0, 0}; c = {0, 0, 1, 0}; d = {0, 0, 0, 1}; e = {2, 0, 0, 0};
f = {0, 2, 0, 0}; g = {0, 0, 2, 0}; h = {0, 0,0, 2};
assoc = Association[1 -> a, 2 -> b, 3 -> c, 4 -> d, 5 -> e, 6 -> f, 7 -> g, 8 -> h]

(*=> <|1 -> {1, 0, 0, 0}, 2 -> {0, 1, 0, 0}, 3 -> {0, 0, 1, 0}, 4 -> {0, 0, 0, 1},
 5 -> {2, 0, 0, 0}, 6 -> {0, 2, 0, 0}, 7 -> {0, 0, 2, 0}, 8 -> {0, 0, 0, 2}|> *)

In general, the association may contain many more terms. You can access the keys, given the values using PositionIndex, e.g.:

PositionIndex[assoc][{2, 0, 0, 0}]
(*=> {5} *)

Suppose the output of a calculation is of the form:

-0.2*{0, 1, 0, 0} + 0.4*{2, 0, 0, 0} + 0.11*{0, 0, 2, 0} + 0.7*{0, 0, 1, 0}

How can you get a list of pairs where the pair consists of the coefficient and the corresponding key of the element of assoc? I.e., for the above output, how can you create the following list?

{{-0.2, 2}, {0.4, 5}, {0.11, 7}, {0.7,3}}

Answer 1

Normally, as @Hausdorff mentioned, you won't get -0.2*{0, 1, 0, 0} + 0.4*{2, 0, 0, 0} + 0.11*{0, 0, 2, 0} + 0.7*{0, 0, 1, 0} as output due to immediate evaluation. But if this is only a simplified example of yours and you're somehow able to preserve the elements in the output, try something like this:

assoc = Association[1 -> a, 2 -> b, 3 -> c, 4 -> d, 5 -> e, 6 -> f, 
   7 -> g, 8 -> h];
iassoc = GeneralUtilities`AssociationInvert[assoc]
Cases[-0.2 b + 0.4*e - 0.11*g + 0.7*c, x_*y_ :> {x, iassoc[y]}]

{{-0.2, 2}, {0.7, 3}, {0.4, 5}, {-0.11, 7}}

(It's recommended you post your original data format so that more specific answers can be given!)