TT* + I is invertible I've the following exercise which I can't solve:
Prove that: 
$$ AA^* + I $$
is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $  A^* $ is the adjoint operator. 
Any help will be appreciated. 

If $ A^*A + I $ is not invertible then $ det(A^*A + I)=0 $ then $-1$ is an eigenvalue of $ A^*A $ then there exist nonzero $v$ such that $A^*Av=-v$ therefore $$ v^*A^*Av=-v^*v$$
which means nonsense because norm is not negative.
Then $ A^*A + I $ is always invertible.
Why does $v$ always imply column-vector and $v^*$ always imply row-vector?
