Question

Problem 3. Recall that a linear map f : V → W is called an isomorphism if it is invertible (i.e. has a linear inverse map). We proved in class that f is in fact invertible if and only if it is bijective. Use this fact from class together with the Rank-Nullity Theorem (of the previous problem) to show that if f : V → V is an endomorphism, then it is actually invertible if

1. it is merely injective

2. it is merely surjective

(you may use the fact that a proper subspace U ⊂ V of a finite dimensional V has strictly smaller dimension than V).

Answer 1