View source for Linear equation

An equation of the form 
$$Ax=b\label{1}$$
where $A$ is a
[[Linear operator|linear operator]] acting from a
[[Vector space|vector space]] $X$ into a vector space $B$, $x$ is an
unknown element of $X$ and $b$ is a given element of $B$ (the free
term). If $b=0$, the linear equation is said to be homogeneous. A
solution of the linear equation is an element $x_0$ that makes (1) an
identity: 
$$Ax\equiv b$$
The simplest example is a linear operator $A:x\mapsto ax$ (a
[[Linear function|linear function]]) and the
[[Linear algebraic equation|linear algebraic equation]] determined by
it: 
$$ax=b,\label{2}$$
$a,b\in \R$ or $\C$ (or an arbitrary field $k$); a solution of it
exists if and only if either $a\ne 0$ (and then $x_0=b/a$) or $a=b=0$ (and then $x_0$
is arbitrary). A generalization of equation (2) is a linear equation
of the form 
$$Ax\equiv f(x)=b,\label{3}$$
where $f(x)$ is a
[[Linear functional|linear functional]] defined on a vector space $X$
over a field $k$ and $b\in k$. In particular, if the dimension of $X$ is
finite and equal to $n$ (so that $X$ is isomorphic to $k^n$), then $f$
is a linear form in several variables $x_1,\dots,x_n=b$ and (3) can be written as

$$a_1x_1+\cdots+a_n x_n = b,\quad a_i,b\in k.\label{4}$$
If the $a_i$ are not all zero simultaneously, then the set of
solutions of (4) forms an $(n-1)$-dimensional linear variety (in the
homogeneous case, a linear subspace) in $X$. If $X$ is infinite
dimensional, then the set of solutions of (3) is a linear variety of
codimension 1.

Several equations of the form (4) constitute a system of linear
equations: 
$$a_{j1}x_1+\cdots+a_{jn} x_n = b_j,\quad j=1,\dots,m.$$
The system (5) can be interpreted as one linear
equation of the form (1) if for $X$ one takes the space $k^n$ and for
$B$ the space $k^m$, and one specifies the operator $A$ by a
[[Matrix|matrix]] $(a_{ij})$, $i=1,\dots,n$, $j=1,\dots,m$. The question of the compatibility of
the system of linear equations (5), that is, the question of the
existence of solutions of the system of linear equations, is settled
by comparing the ranks of the matrices $(a_{ij}$ and $(a_{ij},b)$: there is a
solution if and only if these ranks are equal.

Things are more complicated when $X$ and $B$ are infinite-dimensional
vector spaces. An important role is played by the topologies of the
spaces $X$ and $B$ and various properties of the operator $A$ such as
being bounded, continuous, etc., determined by them. In the general
case the existence and uniqueness of a solution of a linear equation
are determined by the invertibility of $A$ (see
[[Inverse mapping|Inverse mapping]]). However, it is often far from
easy to invert $A$ effectively, and so in the investigation of linear
equations an important role is played by qualitative methods, which
make it possible, without solving the linear equation, to state
properties of the family of solutions (assuming that they exist) that
are useful in certain respects, for example, uniqueness, a priori
estimates, etc. On the other hand, the operator $A$ need not be
defined on the whole space $X$, and equation (1) need not have a
solution for some $b$. In this situation the solvability of (1) is
established (in many practically important cases) by suitably choosing
an extension of $A$ (cf.
[[Extension of an operator|Extension of an operator]]).

For specific types of linear equations, for example for linear
differential equations, both ordinary and partial, and for linear
integral equations, specific methods of solution and investigation,
including numerical ones, have been developed. Finally, in a number of
cases (for example, in linear regression problems) the values of $x_0$,
which are in a certain sense the most suitable for the role of a
solution of the linear equation, turn out to be useful.



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD>
 <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD>
</TR><TR><TD valign="top">[a2]</TD>
 <TD valign="top"> S. Lang, "Linear algebra" , Addison-Wesley (1966)</TD>
</TR><TR><TD valign="top">[a3]</TD>
 <TD valign="top"> P.R. Halmos, "Introduction to Hilbert space and the theory of spectral multiplicity" , Chelsea (1951)</TD>
</TR><TR><TD valign="top">[a4]</TD>
 <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators" , '''1–2''' , Interscience (1958–1959)</TD>
</TR></table>