TITLE: How to show that linear map is invertible?
QUESTION [6 upvotes]: a) Let  $L:V \to V$ be a linear map such that $L^2 + 2L + I = 0$, show that $L$ is invertible.
  b) Let $L:V \to V$ be a linear map such that $L^3 = 0$, show that $I - L$ is invertible.

Here, $I$ is identity mapping.
For first part, I know that $L^2 + 2L + I = (L+I)^2 = 0$, if $v\in V$ then $(L+I)^2 v = (L+I)(L(v) + v)) = 0$ so $L(v) + v$ is in null space of $L+I$, from here how do I show that $0$ is only in null space of $L$.
I don't need exact solution. Hints would suffice.

REPLY [2 votes]: The same proposition deals with both parts:
If $L^n=0$ for some $n\in \mathbb{N}$ then $I-L$ is invertible. To prove this, check that $I+L+L^2 + \cdots + L^{n-1}$ is the inverse. 
a) $(L+I)^2=0$ so $I-(L+I) = -L$ is invertible, and hence so is $L.$ 
b) is precisely the case $n=3.$