linear equationLinearProblem2002-02-22 09:51:472007-03-27 14:12:48Definitionconsistentinconsistentparticular solution\usepackage{amsmath}
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\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{theorem}[proposition]{Theorem}Let $L:U\rightarrow V$ be a linear mapping, and $v\in V$ an element of
the codomain. A {\em linear equation} is
a relation of the form,
$$L(u)=v,$$
where $u\in U$ is to be considered as the unknown. The
solution set of a linear equation is the set of $u\in U$ that satisfy the
above constraint, or to be more precise, the pre-image $L^{-1}(v)$. The equation is
called inconsistent if no solutions exist, that is, if the pre-image is
the empty set. Otherwise, the equation is called \emph{consistent}.
The general solution of
a linear equation has the form
$$u=u_p + u_h,\quad u_p,u_h\in U,$$
where
$$L(u_p)=v$$
is a \emph{particular solution} and where
$$L(u_h)=0$$
is any solution of the corresponding homogeneous problem, i.e. an
element of the kernel of $L$.
{\bf Notes.} Elementary treatments of linear algebra focus almost
exclusively on finite-dimensional linear problems. They neglect to
mention the underlying mapping, preferring to focus instead on
``variables and equations.'' However, the scope of the general concept
is considerably wider, e.g. linear differential equations such as
$$y''+y = 0.$$