How do you prove that a homomorphism is bijective?

Question

so how would you prove that the following homomorphism is bijective so (injective and surjective) and that it is therefore an isomorphism.

T(1 a)(1 b)=T(1 a)+T(1 b)
(0 1)(0 1) (0 1) (0 1)

p.s. the above are matrices

Answer 1

In finite dimension, and endomorphisms,\bijective $\iff$ injective $\iff$ surjective.

Usually the simplest is to prove the endomorphism is injective, i.e. its kernel is $0$.