Let

T:V → Wbe a linear transformation that isbijective, that is, it is

injective1 and surjective2. For eachw ∈ WletT−1(w) be the uniquev ∈ Vsuch that

T(v) =w.vexists becauseTis surjective and it is unique becauseTis injective.T−1 is characterized by the two conditions3 thatT ◦ T−1 = idWandT−1◦ T= idV.

We obtain a mapT−1 :W → V

Show thatT−1 is again a linear transformation.