Identity map isomorphism and addition of isomorphisms

Question

True or false, if $S$ and $T$ be two isomorphisms from $V \to V$,
then so is $S+T$.

The solution provided is $I=S$ and $-I=T$ the identity maps, then $ I + (-I) = 0$

I'm trying to understand how is $-I$ an isomorphism ? I can't seem to picture it , moreover can someone provide me with a different solution so i can strengthen my understanding for this specific problem ? Thank you in advance.

Answer 1

$-I$ is a homomorphism (linear map) because $-I(rv+w)=-(rv+w)=r(-v)+(-w)$, and it is bijective because $-I(v)=-v=0\iff v=0$ and if $v\in V$ then $I(-v))=v$.

Answer 2

Hint:

If you want to prove something is an isomorphism, then you have to check

• It is homomorphism (i.e. it is linear in this case)
• It is bijective (in this case it is enough to check that it kernel is just $\{0\}$).