vector space epimorphism between finite dimensional vector spaces and kernel

Proposition

Let \(f: V \rightarrow W\) be a vectorspace-homomorphism between finite dimensional vector spaces
TFAE:

\(f\) is a vectorspace-epimorphism
\(\mathrm{dim}(\mathrm{ker}(f)) = \mathrm{dim}(V) - \mathrm{dim}(W)\)
\(\mathrm{dim}(\mathrm{ker}(f)) \leq \mathrm{dim}(V) - \mathrm{dim}(W)\)

Proof

1) \(\implies\) 2)

follows from the rank theorem and \(W = \mathrm{im}(f)\):

\begin{align*} \mathrm{dim}(V) =& \mathrm{dim}(\mathrm{ker}(f)) + \mathrm{dim}(\mathrm{im}(f)) \\ \mathrm{dim}(V) =& \mathrm{dim}(\mathrm{ker}(f)) + \mathrm{dim}(W) \\ \end{align*}

2) \(\implies\) 3)

special case

3) \(\implies\) 1)

follows again from the rank theorem and

\begin{align*} \mathrm{dim}(V) =& \mathrm{dim}(\mathrm{ker}(f)) + \mathrm{dim}(\mathrm{im}(f)) \\ \leq& \mathrm{dim}(V) - \mathrm{dim}(W) + \mathrm{dim}(\mathrm{im}(f)) \\ \end{align*}

as this implies

\begin{align*} \mathrm{dim}(W) \leq \mathrm{dim}(\mathrm{im}(f)) \end{align*}

where Dimension eines Unterraums shows

\begin{align*} W = \mathrm{im}(f) \end{align*}