# Theorem Given a [[Linear Map|linear map]] $f$ over [[vector space|vector spaces]] and its [[Adjoint of a Linear Map|adjoint]] $\tilde{f}$, then we have: $ \text{dim}(\text{ker}(f)) - \text{dim}(\text{ker}(\tilde{f})) = \dim{(V)} - \dim{(W)} $ where $\ker$ represents the [[Kernel of a Map|kernel]] and $\dim$ the [[Dimension|dimension]]. # Proof