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سِٽَ 1:
[[Imageعڪس:Secretsharing-3-point.png|thumb|left|ٽِن ڦِرڻَن جي سِڌِر سرشتي سان [[(mathematics)|سَنواٽَن]] جو هڪ مجموعو ملي ٿو. جڏهن ته انٽرسيڪشن جو نقطو انهن مساواتن جو گھربل حل آهي.]]
{{مبسس4}}[[رياضيات]] ۾، '''سڌر مساواتن جو سرشتو''' (يا '''سڌر سرشتو''') ساڳيا ئي [[ڦرڻو(math)|ڦرڻا]] رکندڙ مساواتُن جو مجموعو آهي. مثال طور:
 
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In mathematics, the theory of linear systems is a branch of [[linear algebra]], a subject which is fundamental to modern mathematics. Computational [[algorithm]]s for finding the solutions are an important part of [[numerical linear algebra]], and such methods play a prominent role in [[engineering]], [[physics]], [[chemistry]], [[computer science]], and [[economics]]. A system of non-linear equations can often be [[approximation|approximated]] by a linear system (see [[linearization]]), a helpful technique when making a [[mathematical model]] or [[computer simulation]] of a relatively complex system.
 
== سادو ترين مثال ==
The simplest kind of linear system involves two equations and two variables:
:<math>\begin{alignat}{5}
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This results in a single equation involving only the variable ''y''. Solving gives {{nowrap|''y'' {{=}} 1}}, and substituting this back into the equation for ''x'' yields {{nowrap|''x'' {{=}} 3/2}}. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on [[elementary algebra]].)
 
== عام صورت ==
A general system of ''m'' linear equations with ''n'' unknowns can be written as
:<math>\begin{alignat}{7}
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Often the coefficients and unknowns are [[real number|real]] or [[complex number]]s, but [[integer]]s and [[rational number]]s are also seen, as are polynomials and elements of an abstract [[algebraic structure]].
 
=== طرفڻي مساوات ===
One extremely helpful view is that each unknown is a weight for a [[column vector]] in a [[linear combination]].
:<math>
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\begin{bmatrix}b_1\\b_2\\ \vdots \\b_m\end{bmatrix}
</math>
This allows all the language and theory of ''[[vector space]]s'' (or more generally, ''[[module (mathematics)|modulemodules]]s'') to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the left-hand side is called their ''[[span (linear algebra)|span]]'', and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a ''[[basis (linear algebra)|basis]]'' of [[linearly independent]] vectors that do guarantee exactly one expression; and the number of vectors in that basis (its ''[[dimension (linear algebra)|dimension]]'') cannot be larger than ''m'' or ''n'', but it can be smaller. This is important because if we have ''m'' independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed.
 
=== Matrix equation ===
The vector equation is equivalent to a [[matrix (mathematics)|matrix]] equation of the form
:<math>A\bold{x}=\bold{b}</math>
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The number of vectors in a basis for the span is now expressed as the ''[[rank (linear algebra)|rank]]'' of the matrix.
 
== Solution set ==
[[Imageعڪس:Intersecting Lines.svg|thumb|right|The solution set for the equations {{nowrap|''x'' &minus;− ''y'' {{=}} &minus;1−1}} and {{nowrap|3''x'' + ''y'' {{=}} 9}} is the single point (2,&nbsp;3).]]
A '''solution''' of a linear system is an assignment of values to the variables {{nowrap|''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>''}} such that each of the equations is satisfied. The [[Set (mathematics)|set]] of all possible solutions is called the '''[[solution set]]'''.
 
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For ''n'' variables, each linear equations determines a [[hyperplane]] in [[n-dimensional space|''n''-dimensional space]]. The solution set is the intersection of these hyperplanes, which may be a [[flat (geometry)|flat]] of any dimension.
 
=== General behavior ===
[[Imageعڪس:IntersectingPlanes.png|thumb|right|The solution set for two equations in three variables is usually a line.]]
In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns:
# Usually, a system with fewer equations than unknowns has infinitely many solutions. Such a system is also known as an underdetermined system.
# Usually, a system with the same number of equations and unknowns has a single unique solution.
# Usually, a system with more equations than unknowns has no solution.
In the first case, the [[dimension]] of the solution set is usually equal to {{nowrap|''n'' &minus;− ''m''}}, where ''n'' is the number of variables and ''m'' is the number of equations.
 
The following pictures illustrate this trichotomy in the case of two variables:
:{| border=0 cellpadding=5
|-
|width="150px" align="center"| [[Imageعڪس:One Line.svg|120px]]
|width="150px" align="center"| [[Imageعڪس:Two Lines.svg|120px]]
|width="150px" align="center"| [[Imageعڪس:Three Lines.svg|120px]]
|-
|align="center"| One Equation
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== خاصيتون ==
=== Independence ===
The equations of a linear system are '''independent''' if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as [[linear independence]].
 
[[Imageعڪس:Three Intersecting Lines.svg|thumb|right|The equations {{nowrap|''x'' &minus;− 2''y'' {{=}} &minus;1−1}}, {{nowrap|3''x'' + 5''y'' {{=}} 8}}, and {{nowrap|4''x'' + 3''y'' {{=}} 7}} are not linearly independent.]]
For example, the equations
:<math>3x+2y=6\;\;\;\;\text{and}\;\;\;\;6x+4y=12</math>
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are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.
 
=== Consistency ===
[[Imageعڪس:Parallel Lines.svg|thumb|right|The equations {{nowrap|3''x'' + 2''y'' {{=}} 6}} and {{nowrap|3''x'' + 2''y'' {{=}} 12}} are inconsistent.]]
The equations of a linear system are '''consistent''' if they possess a common solution, and '''inconsistent''' otherwise. When the equations are inconsistent, it is possible to derive a [[contradiction]] from the equations, such as the statement that {{nowrap|0 {{=}} 1}}.
 
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Two linear systems using the same set of variables are '''equivalent''' if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice-versa. Equivalent systems convey precisely the same information about the values of the variables. In particular, two linear systems are equivalent if and only if they have the same solution set.
 
== سِڌِر سرشتي جو حل لهڻ ==
There are several [[algorithm]]s for [[equation solving|solving]] a system of linear equations.
 
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Here ''x'' is the free variable, and ''y'' and ''z'' are dependent.
 
=== Elimination of variables ===
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:
# In the first equation, solve for the one of the variables in terms of the others.
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2x &&\; + \;&& 4y &&\; + \;&& 3z &&\; = \;&& 8 &
\end{alignat}</math>
Solving the first equation for ''x'' gives {{nowrap|''x'' {{=}} 5 + 2''z'' &minus;− 3''y''}}, and plugging this into the second and third equation yields
:<math>\begin{alignat}{5}
-4y &&\; + \;&& 12z &&\; = \;&& -8 & \\
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z &&\; = \;&& 2 && && && && &
\end{alignat}</math>
Substituting {{nowrap|''z'' {{=}} 2}} into the second equation gives {{nowrap|''y'' {{=}} 8}}, and substituting {{nowrap|''z'' {{=}} 2}} and {{nowrap|''y'' {{=}} 8}} into the first equation yields {{nowrap|''x'' {{=}} &minus;15−15}}. Therefore, the solution set is the single point {{nowrap|(''x'', ''y'', ''z'') {{=}} (&minus;15−15, 8, 2)}}.
 
=== Row reduction ===
{{main|Gaussian elimination}}
In '''row reduction''', the linear system is represented as an [[augmented matrix]]:
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0 & 0 & 1 & 2
\end{array}\right].</math>
The last matrix is in reduced row echelon form, and represents the system {{nowrap|''x'' {{=}} &minus;15−15}}, {{nowrap|''y'' {{=}} 8}}, {{nowrap|''z'' {{=}} 2}}. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.
 
=== ڪريمر قاعدو ===
{{main|ڪريمر قاعدو}}
'''Cramer's rule''' is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two [[determinant]]s. For example, the solution to the system
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Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.
 
=== ٻيا طريقا ===
While systems of three or four equations can be readily solved by hand, computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as ''pivoting''. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the [[LU decomposition]] of the matrix ''A''. This is mostly an organizational tool, but it is much quicker if one has to solve several systems with the same matrix ''A'' but different vectors '''b'''.
 
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A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of [[iterative method]]s.
 
== Homogeneous systems ==
A system of linear equations is '''homogeneous''' if all of the constant terms are zero:
:<math>\begin{alignat}{7}
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A homogeneous system is equivalent to a matrix equation of the form
:<math>A\textbf{x}=\textbf{0}</math>
where ''A'' is an {{nowrap|''m'' &times;× ''n''}} matrix, '''x''' is a column vector with ''n'' entries, and '''0''' is the [[zero vector]] with ''m'' entries.
 
=== Solution set ===
Every homogeneous system has at least one solution, known as the '''zero solution''' (or '''trivial solution'''), which is obtained by assigning the value of zero to each of the variables. The solution set has the following additional properties:
# If '''u''' and '''v''' are two [[vector (mathematics)|vectors]] representing solutions to a homogeneous system, then the vector sum {{nowrap|'''u''' + '''v'''}} is also a solution to the system.
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These are exactly the properties required for the solution set to be a [[Euclidean subspace|linear subspace]] of '''R'''<sup>''n''</sup>. In particular, the solution set to a homogeneous system is the same as the [[Kernel (matrix)|null space]] of the corresponding matrix ''A''.
 
=== Relation to nonhomogeneous systems ===
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
:<math>A\textbf{x}=\textbf{b}\qquad \text{and}\qquad A\textbf{x}=\textbf{0}\text{.}</math>
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* [[Matrix decomposition]]
 
== Notes ==
{{reflist}}
 
== حوالا ==
{{see also|Linear algebra#Further reading}}
 
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* [http://www.youtube.com/watch?v=gVMRuLH6FdQ Lec 1| 18.06 Linear Algebra, Spring 2005], (W. Gilbert Strang), School: MIT
 
[[Categoryزمرو:Equations]]
[[Categoryزمرو:Linear algebra]]
 
[[ar:نظام المعادلات الخطية]]
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[[cs:Soustava lineárních rovnic]]
[[de:Lineares Gleichungssystem]]
[[et:Lineaarvõrrandisüsteem]]▼
[[el:Σύστημα γραμμικών εξισώσεων]]
[[esen:SistemaSystem deof ecuacioneslinear linealesequations]]
[[eo:Sistemo de linearaj ekvacioj]]
[[es:Sistema de ecuaciones lineales]]
▲[[et:Lineaarvõrrandisüsteem]]
[[fa:دستگاه معادلات خطی]]
[[fi:Lineaarinen yhtälöryhmä]]▼
[[fr:Système d'équations linéaires]]
[[ko:일차연립방정식]]▼
[[hr:Sustav linearnih jednadžbi]]
[[is:Línulegt jöfnuhneppi]]
[[it:Sistema di equazioni lineari]]
[[nl:Stelsel van lineaire vergelijkingen]]▼
[[ja:線型方程式系]]
▲[[ko:일차연립방정식]]
▲[[nl:Stelsel van lineaire vergelijkingen]]
[[pl:Układ równań liniowych]]
[[pt:Sistema de equações lineares]]
[[ru:Система линейных алгебраических уравнений]]
[[sr:Систем линеарних једначина]]
▲[[fi:Lineaarinen yhtälöryhmä]]
[[sv:Linjärt ekvationssystem]]
[[uk:Система лінійних алгебраїчних рівнянь]]
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