homogeneous linear problem HomogeneousLinearProblem 2002-02-22 10:04:14 2007-03-27 16:00:06 Definition linear equation \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} Let $L:U\rightarrow V$ be a linear mapping. A linear equation is called {\em homogeneous} if it has the form $$L(u)=0,\quad u\in U.$$ A homogeneous linear problem always has a trivial solution, namely $u=0$. The key issue in homogeneous problems is, therefore, the question of the existence of non-trivial solutions, i.e. whether or not the kernel of $L$ is trivial, or equivalently, whether or not $L$ is injective.