Let
T : V → W be a linear transformation that is bijective, that is, it is
injective1 and surjective2. For each w ∈ W let T−1(w) be the unique v ∈ V such that
T(v) = w. v exists because T is surjective and it is unique because T is injective. T−1 is characterized by the two conditions3 that T ◦ T−1 = idW and T−1 ◦ T = idV .
We obtain a map T−1 : W → V
Show that T−1 is again a linear transformation.