Revision as of 00:30, 20 April 2025 by Bot (Created page with "<div class="d-none"><math> \newcommand{\mathds}{\mathbb}</math></div>Clarify the theory of bases and dimensions for the linear subspaces <math>V\subset\mathbb R^N</math>, notably by establishing the formula <math display="block"> \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>.")
BBot
Apr 20'25
Exercise
[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].