[math] \newcommand{\mathds}{\mathbb}[/math]

Clarify the theory of bases and dimensions for the linear subspaces [math]V\subset\mathbb R^N[/math], notably by establishing the formula

[[math]] \dim(\ker f)+\dim(Im f)=N [[/math]]

valid for any linear map [math]f:\mathbb C^N\to\mathbb C^N[/math], and then extend this into a theory of abstract linear spaces [math]V[/math], which are not necessarily subspaces of [math]\mathbb C^N[/math].