Pensieve header: Some utilities for linear algebra (passe).

LC2SA is “Linear Combination(s) to Sparse Array”.

LC2SA[bas_List, lc_List] := Module[
{a1, a2, rule},
rule = Thread[bas -> Array[b, {Length[bas]}]];
SparseArray[
Flatten[Table[
Cases[List @@ (a1+a2+lc[[k]] /. rule), c_.*b[i_] :> ({k,i} -> c)],
{k, Length[lc]}
]],
{Length[lc], Length[bas]}
]
];
LC2SA[bas_List, lc_] := SparseArray[1->1].LC2SA[bas, {lc}];

LC2SA is superseded by CoefficientArray:

?CoefficientArrays