linear equation

Let $L:U\rightarrow V$ be a linear mapping, and $v\in V$ an element of the codomain. A 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 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 particular solution and where $$L(u_h)=0$$ is any solution of the corresponding homogeneous problem, i.e. an element of the kernel of $L$ .

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 $$y''+y = 0.$$