Generating a vector basis

Question

I have several lists parameterizing a vector space, for instance

{a[1],a[2],a[1]+2 a[2]-a[3]}

For each list I want to generate a basis such as {{1,0,1},{0,1,2},{0,0,-1}} so that

{a[1],a[2],a[1]+2 a[2]-a[3]} == a[1] {1,0,1} + a[2] {0,1,2 } + a[3] {0,0,-1 } 

The function Position[] does not look helpful for this.

Answer 1

Try something like this:

Map[Coefficient[{a[1], a[2], a[1] + 2 a[2] - a[3]}, #] &, Table[a[i], {i, 3}]]
(*{{1, 0, 1}, {0, 1, 2}, {0, 0, -1}}*)

Answer 2

CoefficientArrays[] does nicely for this:

CoefficientArrays[{a[1], a[2], a[1] + 2 a[2] - a[3]}, Array[a, 3]] // 
   Normal // Last // Transpose

{{1, 0, 1}, {0, 1, 2}, {0, 0, -1}}

Answer 3

This is very short:

vec = {a[1], a[2], a[1] + 2 a[2] - a[3]};

Transpose[D[vec, {Array[a, 3]}]]

(* ==> {{1, 0, 1}, {0, 1, 2}, {0, 0, -1}} *)

It gets the basis from the columns of the Jacobian for the linear relation in vec.