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An equation of the form
+
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:
 +
$$Ax\cong b$$
 +
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
 +
$$Ax\cong f(x)=b,\label{3}$$
 +
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$a_1x_1+\cdots+a_n x_n = b,\quad a_i,b\in k.\label{4}$$
 +
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591902.png" /> is a [[Linear operator|linear operator]] acting from a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591903.png" /> into a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591905.png" /> is an unknown element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591907.png" /> is a given element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591908.png" /> (the free term). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l0591909.png" />, the linear equation is said to be homogeneous. A solution of the linear equation is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919010.png" /> that makes (1) an identity:
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919011.png" /></td> </tr></table>
+
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]]).
  
The simplest example is a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919012.png" /> (a [[Linear function|linear function]]) and the [[Linear algebraic equation|linear algebraic equation]] determined by it:
+
For specific types of linear equations, for example for linear
 
+
differential equations, both ordinary and partial, and for linear
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
integral equations, specific methods of solution and investigation,
 
+
including numerical ones, have been developed. Finally, in a number of
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919015.png" /> (or an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919016.png" />); a solution of it exists if and only if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919017.png" /> (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919018.png" />) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919019.png" /> (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919020.png" /> is arbitrary). A generalization of equation (2) is a linear equation of the form
+
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
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
solution of the linear equation, turn out to be useful.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919022.png" /> is a [[Linear functional|linear functional]] defined on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919023.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919025.png" />. In particular, if the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919026.png" /> is finite and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919027.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919028.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919029.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919030.png" /> is a linear form in several variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919031.png" /> and (3) can be written as
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
 
 
 
If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919033.png" /> are not all zero simultaneously, then the set of solutions of (4) forms an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919034.png" />-dimensional linear variety (in the homogeneous case, a linear subspace) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919036.png" /> 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:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
 
 
 
The system (5) can be interpreted as one linear equation of the form (1) if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919038.png" /> one takes the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919039.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919040.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919041.png" />, and one specifies the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919042.png" /> by a [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919045.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919047.png" />: there is a solution if and only if these ranks are equal.
 
 
 
Things are more complicated when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919049.png" /> are infinite-dimensional vector spaces. An important role is played by the topologies of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919051.png" /> and various properties of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919052.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919053.png" /> (see [[Inverse mapping|Inverse mapping]]). However, it is often far from easy to invert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919054.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919055.png" /> need not be defined on the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919056.png" />, and equation (1) need not have a solution for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919057.png" />. In this situation the solvability of (1) is established (in many practically important cases) by suitably choosing an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919058.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059190/l05919059.png" />, which are in a certain sense the most suitable for the role of a solution of the linear equation, turn out to be useful.
 
  
  
Line 37: Line 77:
  
 
====References====
 
====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>
+
<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>

Revision as of 14:08, 17 November 2011

An equation of the form $$Ax=b\label{1}$$ where $A$ is a linear operator acting from a 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: $$Ax\cong b$$ The simplest example is a linear operator $A:x\mapsto ax$ (a linear function) and the 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 $$Ax\cong f(x)=b,\label{3}$$ where $f(x)$ is a 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 $(a_{ij})$, $i=1,\dots,n$, $j=1,\dots,m$. 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). 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).

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

[a1] N. Bourbaki, "Elements of mathematics. Algebra:

Algebraic structures. Linear algebra" , 1 , Addison-Wesley

(1974) pp. Chapt.1;2 (Translated from French)
[a2] S. Lang, "Linear algebra" , Addison-Wesley (1966)
[a3] P.R. Halmos, "Introduction to Hilbert space and the theory of spectral multiplicity" , Chelsea (1951)
[a4] N. Dunford, J.T. Schwartz,

"Linear operators" , 1–2 , Interscience

(1958–1959)
How to Cite This Entry:
Linear equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_equation&oldid=11214
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article