(I will phrase the question in terms of $\mathbb{R}^n$)
Is the following statement a standard well-known linear algebra fact that I can quote without proving? (Perhaps more importantly, is it even true?)
Notation that we will use:
$a_M(\lambda) =$ algebraic multiplicity of $\lambda$ for matrix/ linear map $M$.
$g_M(\lambda) =$ same for geometric multiplicity.
Statement:
Let $A \in \mathbb{R}^{n \times n}$. Then
$\mathbb{R}^n = \textrm{ker}A \oplus \textrm{range}A \Leftrightarrow a_A(0) = g_A(0).$
Thanks,
Julian.