Bijective Linear Transformation Proof

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.

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Originally Posted by amm345

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.

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Let , so:

As ....take it from here, and very simmilar with multiplication by a scalar.

Tonio