Question

1.a Implication: If an endomorphism f : V → V is not an isomorphism, then it must have a non-trivial kernel. Use this implication, together with the definition of determinant given in class, to show that if for an endomorphism f : V → V we have det(f) 6 /= 0, then in fact f is an isomorphism.

1.b.

The Rank-Nullity Theorem applies to a linear map f : V → W (where V is finite dimensional) and claims that: dim(V) = dim(im(f)) + dim(ker(f)) = "rank" + "nullity"

Write a brief proof of the Rank-Nullity Theorem using the First Isomorphism Theorem together with the following numerical fact: dim(V/U) = dim(V) − dim(U) for U ⊂ V a subspace and V/U the quotient.

Answer 1