Rank-nullity Theorem: From Some basic homological algebra and first isomorphism theorem

 

 

 

Consider the following digram in ${\mathrm{Vect}}_{\mathbb{F}}$$\mathrm{Vect}_{\mathbb F}$.

Since free module is projective, hence we have the exact sequence split, i.e. ${\mathbb{R}}^n\cong {\mathbb{R}}^k\oplus {\mathbb{R}}^m$$\R^n\cong \R^k\oplus\mathbb R^m$.

Hence we have $n=m+k$$n=m+k$. i.e. $\dim V/K+\dim K=\dim V$$\dim V/K+\dim K=\dim V$. By the first isomorphism theorem, it is

$$\begin{array}{}\text{(1)}& \dim \mathrm{Im}(g)+\dim \ker (g)=\dim V\end{array}$$