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System of linear equations
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A linear system in three variables determines a collection of planes. The intersection point is the solution.
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables.[1] For example,
3 x + 2 y − z = 1 2 x − 2 y + 4 z = − 2 − x + 1 2 y − z = 0 {\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by
x = 1 y = − 2 z = − 2 {\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}
since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.
In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.
Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.
Contents
[hide]
1 Elementary example
2 General form
2.1 Vector equation
2.2 Matrix equation
3 Solution set
3.1 Geometric interpretation
3.2 General behavior
4 Properties
4.1 Independence
4.2 Consistency
4.3 Equivalence
5 Solving a linear system
5.1 Describing the solution
5.2 Elimination of variables
5.3 Row reduction
5.4 Cramer's rule
5.5 Matrix solution
5.6 Other methods
6 Homogeneous systems
6.1 Solution set
6.2 Relation to nonhomogeneous systems
7 See also
8 Notes
9 References
9.1 Textbooks
Elementary example[edit]
The simplest kind of linear system involves two equations and two variables:
2 x + 3 y = 6 4 x + 9 y = 15 . {\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}
One method for solving such a system is as follows. First, solve the top equation for x {\displaystyle x} in terms of y {\displaystyle y} :
x = 3 − 3 2 y . {\displaystyle x=3-{\frac {3}{2}}y.}
Now substitute this expression for x into the bottom equation:
4 ( 3 − 3 2 y ) + 9 y = 15. {\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}
This results in a single equation involving only the variable y {\displaystyle y} . Solving gives y = 1 {\displaystyle y=1} , and substituting this back into the equation for x {\displaystyle x} yields x = 3 / 2 {\displaystyle x=3/2} . This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra.)
General form[edit]
A general system of m linear equations with n unknowns can be written as
a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m , {\displaystyle {\begin{aligned}a_{11}x_{1}&+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}&+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots &\\a_{m1}x_{1}&+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m},\end{aligned}}}
where x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} are the unknowns, a 11 , a 12 , … , a m n {\displaystyle a_{11},a_{12},\ldots ,a_{mn}} are the coefficients of the system, and b 1 , b 2 , … , b m {\displaystyle b_{1},b_{2},\ldots ,b_{m}} are the constant terms.
Often the coefficients and unknowns are real or complex numbers, but integers and rational numbers are also seen, as are polynomials and elements of an abstract algebraic structure.
Vector equation[edit]
One extremely helpful view is that each unknown is a weight for a column vector in a linear combination.
x 1 [ a 11 a 21 ⋮ a m 1 ] + x 2 [ a 12 a 22 ⋮ a m 2 ] + ⋯ + x n [ a 1 n a 2 n ⋮ a m n ] = [ b 1 b 2 ⋮ b m ] {\displaystyle x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\cdots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}
This allows all the language and theory of vector spaces (or more generally, modules) 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, 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 of linearly independent vectors that do guarantee exactly one expression; and the number of vectors in that basis (its 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[edit]
The vector equation is equivalent to a matrix equation of the form
A x = b {\displaystyle A{\mathbf {x}}={\mathbf {b}}}
where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.
A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] , x = [ x 1 x 2 ⋮ x n ] , b = [ b 1 b 2 ⋮ b m ] {\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad {\mathbf {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad {\mathbf {b}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}
The number of vectors in a basis for the span is now expressed as the rank of the matrix.
Solution set[edit]
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3).
A solution of a linear system is an assignment of values to the variables x1, x2, ..., xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set.
A linear system may behave in any one of three possible ways:
The system has infinitely many solutions.
The system has a single unique solution.
The system has no solution.
Geometric interpretation[edit]
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.
For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.[2]
For n variables, each linear equation determines a hyperplane in n-dimensional space. The solution set is the intersection of these hyperplanes, and is a flat, which may have any dimension lower than n.
General behavior[edit]
The solution set for two equations in three variables is, in general, 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. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations.
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system.
In general, a system with the same number of equations and unknowns has a single unique solution.
In general, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.
In the first case, the dimension of the solution set is, in general, equal to n − 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:
One equation Two equations Three equations
The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.
It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).
A system of linear equations behave differently from the general case if the equations are linearly dependent, or if it is inconsistent and has no more equations than unknowns.
Properties[edit]
Independence[edit]
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.
The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent.
For example, the equations
3 x + 2 y = 6 and 6 x + 4 y = 12 {\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=12}
are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.
For a more complicated example, the equations
x − 2 y = − 1 3 x + 5 y = 8 4 x + 3 y = 7 {\displaystyle {\begin{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}
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[edit]
See also: Consistent and inconsistent equations
The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent.
A linear system is inconsistent if it has no solution, and otherwise it is said to be consistent. When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten as the statement 0 = 1.
For example, the equations
3 x + 2 y = 6 and 3 x + 2 y = 12 {\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;3x+2y=12}
are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get 0 = 1. The graphs of these equations on the xy-plane are a pair of parallel lines.
It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
x + y = 1 2 x + y = 1 3 x + 2 y = 3 {\displaystyle {\begin{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}
are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1. Note that any two of these equations have a common solution. The same phenomenon can occur for any number of equations.
In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.
Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions. The rank of a system of equations (i.e. the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [the number of variables] + 1.
Equivalence[edit]
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. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.
Solving a linear system[edit]
There are several algorithms for solving a system of linear equations.
Describing the solution[edit]
When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, for example ( x = 3 , y = − 2 , z = 6 ) {\displaystyle (x=3,\;y=-2,\;z=6)} . When an order on the unknowns has been fixed, for example the alphabetical order the solution may be described as a vector of values, like ( 3 , − 2 , 6 ) {\displaystyle (3,\,-2,\,6)} for the previous example.
To describe a set with an infinite number of solutions, typically some of the variables are designated as free (or independent, or as parameters), meaning that they are allowed to take any value, while the remaining variables are dependent on the values of the free variables.
For example, consider the following system:
x + 3 y − 2 z = 5 3 x + 5 y + 6 z = 7 {\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}
The solution set to this system can be described by the following equations:
x = − 7 z − 1 and y = 3 z + 2 . {\displaystyle x=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}
Here z is the free variable, while x and y are dependent on z. Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y.
Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a plane, or higher-dimensional set.
Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows:
y = − 3 7 x + 11 7 and z = − 1 7 x − 1 7 . {\displaystyle y=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}
Here x is the free variable, and y and z are dependent.
Elimination of variables[edit]
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 one of the variables in terms of the others.
Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and one fewer unknown.
Repeat until the system is reduced to a single linear equation.
Solve this equation, and then back-substitute until the entire solution is found.
For example, consider the following system:
x + 3 y − 2 z = 5 3 x + 5 y + 6 z = 7 2 x + 4 y + 3 z = 8 {\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\\2x&&\;+\;&&4y&&\;+\;&&3z&&\;=\;&&8&\end{alignedat}}}
Solving the first equation for x gives x = 5 + 2z − 3y, and plugging this into the second and third equation yields
− 4 y + 12 z = − 8 − 2 y + 7 z = − 2 {\displaystyle {\begin{alignedat}{5}-4y&&\;+\;&&12z&&\;=\;&&-8&\\-2y&&\;+\;&&7z&&\;=\;&&-2&\end{alignedat}}}
Solving the first of these equations for y yields y = 2 + 3z, and plugging this into the second equation yields z = 2. We now have:
x = 5 + 2 z − 3 y y = 2 + 3 z z = 2 {\displaystyle {\begin{alignedat}{7}x&&\;=\;&&5&&\;+\;&&2z&&\;-\;&&3y&\\y&&\;=\;&&2&&\;+\;&&3z&&&&&\\z&&\;=\;&&2&&&&&&&&&\end{alignedat}}}
Substituting z = 2 into the second equation gives y = 8, and substituting z = 2 and y = 8 into the first equation yields x = −15. Therefore, the solution set is the single point (x, y, z) = (−15, 8, 2).
Row reduction[edit]
Main article: Gaussian elimination
In row reduction (also known as Gaussian elimination), the linear system is represented as an augmented matrix:
[ 1 3 − 2 5 3 5 6 7 2 4 3 8 ] . {\displaystyle \left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]{\text{.}}}
This matrix is then modified using elementary row operations until it reaches reduced row echelon form. There are three types of elementary row operations:
Type 1: Swap the positions of two rows.
Type 2: Multiply a row by a nonzero scalar.
Type 3: Add to one row a scalar multiple of another.
Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination. The following computation shows Gauss-Jordan elimination applied to the matrix above:
[ 1 3 − 2 5 3 5 6 7 2 4 3 8 ] ∼ [ 1 3 − 2 5 0 − 4 12 − 8 2 4 3 8 ] ∼ [ 1 3 − 2 5 0 − 4 12 − 8 0 − 2 7 − 2 ] ∼ [ 1 3 − 2 5 0 1 − 3 2 0 − 2 7 − 2 ] ∼ [ 1 3 − 2 5 0 1 − 3 2 0 0 1 2 ] ∼ [ 1 3 − 2 5 0 1 0 8 0 0 1 2 ] ∼ [ 1 3 0 9 0 1 0 8 0 0 1 2 ] ∼ [ 1 0 0 − 15 0 1 0 8 0 0 1 2 ] . {\displaystyle {\begin{aligned}\left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\2&4&3&8\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\0&-2&7&-2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&-2&7&-2\end{array}}\right]\\&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&0&9\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&0&0&-15\\0&1&0&8\\0&0&1&2\end{array}}\right].\end{aligned}}}
The last matrix is in reduced row echelon form, and represents the system x = −15, y = 8, 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.
Cramer's rule[edit]
Main article: Cramer's rule
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 determinants. For example, the solution to the system
x + 3 y − 2 z = 5 3 x + 5 y + 6 z = 7 2 x + 4 y + 3 z = 8 {\displaystyle {\begin{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}
is given by
x = | 5 3 − 2 7 5 6 8 4 3 | | 1 3 − 2 3 5 6 2 4 3 | , y = | 1 5 − 2 3 7 6 2 8 3 | | 1 3 − 2 3 5 6 2 4 3 | , z = | 1 3 5 3 5 7 2 4 8 | | 1 3 − 2 3 5 6 2 4 3 | . {\displaystyle x={\frac {\,\left|{\begin{matrix}5&3&-2\\7&5&6\\8&4&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;y={\frac {\,\left|{\begin{matrix}1&5&-2\\3&7&6\\2&8&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;z={\frac {\,\left|{\begin{matrix}1&3&5\\3&5&7\\2&4&8\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}}.}
For each variable, the denominator is the determinant of the matrix of coefficients, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.
Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.) 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.[citation needed]
Matrix solution[edit]
If the equation system is expressed in the matrix form A x = b {\displaystyle A{\mathbf {x}}={\mathbf {b}}} , the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution given by
x = A − 1 b {\displaystyle {\mathbf {x}}=A^{-1}{\mathbf {b}}}
where A − 1 {\displaystyle A^{-1}} is the inverse of A. More generally, regardless of whether m=n or not and regardless of the rank of A, all solutions (if any exist) are given using the Moore-Penrose pseudoinverse of A, denoted A † {\displaystyle A^{\dagger }} , as follows:
x = A † b + ( I − A † A ) w {\displaystyle {\mathbf {x}}=A^{\dagger }{\mathbf {b}}+(I-A^{\dagger }A){\mathbf {w}}}
where w {\displaystyle {\mathbf {w}}} is a vector of free parameters that ranges over all possible n×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using w = 0 {\displaystyle {\mathbf {w}}={\mathbf {0}}} satisfy A x = b {\displaystyle A{\mathbf {x}}={\mathbf {b}}} — that is, that A A † b = b . {\displaystyle AA^{\dagger }{\mathbf {b}}={\mathbf {b}}.} If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which A is square and of full rank, A † {\displaystyle A^{\dagger }} simply equals A − 1 {\displaystyle A^{-1}} and the general solution equation simplifies to x = A − 1 b + ( I − A − 1 A ) w = A − 1 b + ( I − I ) w = A − 1 b {\displaystyle {\mathbf {x}}=A^{-1}{\mathbf {b}}+(I-A^{-1}A){\mathbf {w}}=A^{-1}{\mathbf {b}}+(I-I){\mathbf {w}}=A^{-1}{\mathbf {b}}} as previously stated, where w {\displaystyle {\mathbf {w}}} has completely dropped out of the solution, leaving only a single solution. In other cases, though, w {\displaystyle {\mathbf {w}}} remains and hence an infinitude of potential values of the free parameter vector w {\displaystyle {\mathbf {w}}} give an infinitude of solutions of the equation.
Other methods[edit]
While systems of three or four equations can be readily solved by hand (see Cracovian), 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.
If the matrix A has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition. Levinson recursion is a fast method for Toeplitz matrices. Special methods exist also for matrices with many zero elements (so-called sparse matrices), which appear often in applications.
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 methods.
Homogeneous systems[edit]
See also: Homogeneous differential equation
A system of linear equations is homogeneous if all of the constant terms are zero:
a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = 0 ⋮ ⋮ ⋮ ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = 0. {\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\,\vdots \\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}
A homogeneous system is equivalent to a matrix equation of the form
A x = 0 {\displaystyle A{\textbf {x}}={\textbf {0}}}
where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.
Solution set[edit]
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. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties:
If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system.
If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system.
These are exactly the properties required for the solution set to be a linear subspace of Rn. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A. Numerical solutions to a homogeneous system can be found with a singular value decomposition.
Relation to nonhomogeneous systems[edit]
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
A x = b and A x = 0 . {\displaystyle A{\textbf {x}}={\textbf {b}}\qquad {\text{and}}\qquad A{\textbf {x}}={\textbf {0}}{\text{.}}}
Specifically, if p is any specific solution to the linear system Ax = b, then the entire solution set can be described as
{ p + v : v is any solution to A x = 0 } . {\displaystyle \left\{{\textbf {p}}+{\textbf {v}}:{\textbf {v}}{\text{ is any solution to }}A{\textbf {x}}={\textbf {0}}\right\}.}
Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector p.
This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.
See also[edit]
Arrangement of hyperplanes
Iterative refinement
Coates graph
LAPACK (the free standard package to solve linear equations numerically; available in Fortran, C, C++)
Linear equation over a ring
Linear least squares
Matrix decomposition
Matrix splitting
NAG Numerical Library (NAG Library versions of LAPACK solvers)
Simultaneous equations
Moore–Penrose pseudoinverse
Notes[edit]
Jump up ^ The subject of this article is basic in mathematics, and is treated in many textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.
Jump up ^ Charles G. Cullen (1990). Matrices and Linear Transformations. MA: Dover. p. 3. ISBN 978-0-486-66328-9.
References[edit]
See also: Linear algebra § Further reading
Textbooks[edit]
Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0
Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7
Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8
Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3
Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
Strang, Gilbert (2005), Linear Algebra and Its Applications
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Where Are You Going, Where Have You Been?
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"Where Are You Going, Where Have You Been?" is a frequently anthologized short story written by Joyce Carol Oates. The story first appeared in the Fall 1966 edition of Epoch magazine. It was inspired by three Tucson, Arizona murders committed by Charles Schmid, which were profiled in Life magazine in an article written by Don Moser on March 4, 1966.[1] Oates said that she dedicated the story to Bob Dylan because she was inspired to write it after listening to his song "It's All Over Now, Baby Blue".[2][3]
Contents
[hide]
1 Plot
2 Critical review
3 Adaptations
4 References
5 External links
Plot[edit]
The story was originally named "Death and the Maiden".[citation needed] The main character of Oates' story is Connie, a beautiful, self-absorbed 15-year-old girl, who is at odds with her mother—once a beauty herself—and with her dutiful, "steady", and homely older sister. Without her parents' knowledge, she spends most of her evenings picking up boys at a Big Boy restaurant, and one evening captures the attention of a stranger in a gold convertible covered with cryptic writing. While her parents are away at her aunt's barbecue, two men pull up in front of Connie's house and call her out. She recognizes the driver, Arnold Friend, as the man from the drive-in restaurant, and is initially charmed by the smooth-talking, charismatic stranger. He tells Connie he is 18 and has come to take her for a ride in his car with his sidekick Ellie. Connie slowly realizes that he is actually much older,[4] and grows afraid. When she refuses to go with them, Friend becomes more forceful and threatening, saying that he will harm her family, while at the same time appealing to her vanity, saying that she is too good for them. Connie is compelled to leave with him and do what he demands of her.
Critical review[edit]
Considerable academic analysis has been written about the story, with scholars divided on whether it is intended to be taken literally or as allegory. Several writers focus on the series of numbers written on Friend's car, which he indicates are a code of some sort, but which is never explained:
"Now, these numbers are a secret code, honey," Arnold Friend explained. He read off the numbers 33, 19, 17 and raised his eyebrows at her to see what she thought of that, but she didn't think much of it.
Literary scholars have interpreted this series of numbers as different Biblical references,[4][5] as an underlining of Friend's sexual deviancy,[6] or as a reference to the ages of Friend and his victims.[4]
The narrative has also been viewed as an allegory for initiation into sexual adulthood,[7] an encounter with the devil, a critique of modern youth's obsession with sexual themes in popular music,[8] or as a dream sequence.[9]
Adaptations[edit]
The story was loosely adapted into the 1985 film Smooth Talk, starring Laura Dern and Treat Williams.[10] Oates has written an essay named "Smooth Talk: Short Story into Film" about the adaptation.[11]
The short story is the inspiration and basis for The Blood Brothers' song "The Salesman, Denver Max".
References[edit]
Jump up ^ The Pied Piper of Tucson by Don & Jerry Cohen Moser | Kirkus Reviews. ISBN 978-0453001243.
Jump up ^ Oates, J.C. & Showalter, E. (1994). "Introduction". Where Are You Going, Where Have You Been. Rutgers University Press. p. 9. ISBN 0-8135-2135-1.
Jump up ^ Where Are You Going, Where Have You Been and Bob Dylan, 2011-01-11, retrieved 2018-04-24
^ Jump up to: a b c Pinewski, David (Spring 1991). "Oates's 'Where Are You Going, Where Have You Been?'". Explicator. 49 (3): 195–196. |access-date= requires |url= (help)
Jump up ^ Robson, Mark (Summer 1982). "Oates's 'Where Are You Going, Where Have You Been?'". The Explicator. 40: 59–60.
Jump up ^ Hurley, C. Harold (Winter 1987). "Cracking the Secret Code in Oats's 'Where Are You Going, Where Have You Been'". Studies in Short Fiction. 24 (1): 62–66.
Jump up ^ Urbanski, Marie Mitchell Olesen (1978). "Existential Allegory: Joyce Carol Oates's 'Where Are You Going, Where Have You Been?'". Studies in Short Fiction. 15: 200–203. |access-date= requires |url= (help)
Jump up ^ Petry, Alice Hall (Spring 1988). "Who Is Ellie? Oates's 'Where Are You Going, Where Have You Been?'". Studies in Short Fiction. 25 (2): 155–157. |access-date= requires |url= (help)
Jump up ^ Rubin, Larry (Summer 1984). "Oates's 'Where Are You Going, Where Have You Been?'". Explicator. 42 (4): 57–59. |access-date= requires |url= (help)
Jump up ^ Dickinson, P. (July 2008). "Riding in Cars with Boys: Reconsidering 'Smooth Talk'". Literature Film Quarterly. 36 (3): 202–214. |access-date= requires |url= (help)
Jump up ^ Oates, Joyce Carol (2016-10-10). ""Where Are You Going, Where Have You Been?" and Smooth Talk: Short Story Into Film". Celestial Timepiece. Retrieved 2018-04-24.
External links[edit]
Complete text on Celestial Timepiece, an authorized Joyce Carol Oates Home Page
Retrieved from "https://en.wikipedia.org/w/index.php?title=Where_Are_You_Going,_Where_Have_You_Been%3F&oldid=837981099"
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The Young and the Restless cast members
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The Young and the Restless is an American television soap opera, created by William J. Bell and Lee Phillip Bell for CBS. It first aired on March 26, 1973. The longest-running current cast member is Doug Davidson, who has portrayed private investigator Paul Williams since May 23, 1978.[1] Jeanne Cooper, who portrayed the soap opera's matriarch Katherine Chancellor, holds the record for the series' longest-running cast member, airing from November 1973 until her death in May 2013.[2] Melody Thomas Scott and Eric Braeden, who portray Nikki and Victor Newman, are the second and third longest-running cast members, having joined in February 1979 and February 1980, respectively.[1][3] Kate Linder has portrayed Esther Valentine since April 1982, and rounds out the series' top four longest-running cast members.[4] The following list is of cast members who are currently on the show: both main and recurring members, as well as those who are debuting, departing or returning from the series.
Contents
[hide]
1 Cast
1.1 Main cast
1.2 Recurring cast
2 Previous cast members
3 References and notes
4 External links
Cast
Main cast
Actor
Character
Duration
Peter Bergman
Jack Abbott 1989–
Marco Annicelli 2015
Eric Braeden
Victor Newman 1980–
Sharon Case
Sharon Newman 1994–
Doug Davidson
Paul Williams 1978–
Eileen Davidson
Ashley Abbott 1982–88, 1999–
Melissa Claire Egan
Chelsea Newman 2011–
Cait Fairbanks
Tessa Porter 2017–
Noah Alexander Gerry
Charlie Ashby 2017–
Daniel Goddard
Cane Ashby 2007–
Caleb Atkinson 2011
Camryn Grimes
Cassie Newman 1997–2007, 2009–10, 2013–14
Mariah Copeland 2014–
Daniel Hall
Scott Grainger 2017–
Amelia Heinle
Victoria Newman 2005–
Bryton James
Devon Hamilton 2004–
Christel Khalil
Lily Ashby 2002–
Tristan Lake Leabu
Reed Hellstrom 2016–
Christian LeBlanc
Michael Baldwin 1991–93, 1997–
Kate Linder
Esther Valentine 1982–
Thad Luckinbill
J.T. Hellstrom 1999–2010, 2017–
Mishael Morgan
Hilary Hamilton 2013–
Joshua Morrow
Nicholas Newman 1994–
Melissa Ordway
Abby Newman 2013–
Melody Thomas Scott
Nikki Newman 1979–
Kristoff St. John
Neil Winters 1991–
Lexie Stevenson
Mattie Ashby 2017–
Jason Thompson
Billy Abbott 2016–
Gina Tognoni
Phyllis Summers 2014–
Recurring cast
Actor
Character
Duration
Marla Adams
Dina Abbott Mergeron 1983–86, 1991, 1996, 2008, 2017–
Robert Adamson
Noah Newman 2012–
Catherine Bach
Anita Lawson 2012–
Lauralee Bell
Christine Blair Williams 1983–2006, 2010–
Corbin Bernsen
Todd Williams 2003–04, 2009–10, 2012–14, 2017–
Tracey E. Bregman
Lauren Fenmore Baldwin 1983–95, 2000–
Sarah Smythe 2010
Sean Carrigan
Stitch Rayburn 2013–
Judith Chapman
Jill Foster Abbott 1994
Gloria Abbott Bardwell 2005–
Tamara Clatterbuck
Alice Johnson 1998–2000, 2005, 2017–
Annalisa Cochrane
Zoey 2017–
Angell Conwell
Leslie Michaelson 2010–
Jerry Douglas
John Abbott 1982–2013, 2015–
Alistair Wallingford 2008
Max Ehrich
Fenmore Baldwin 2012–
Margarita Franco
Mrs. Martinez 2016–
Kelly Frye
Michelle Hazelton 2016–
David Faustino
Howard Green 2016–
Robert Gant
David Sherman 2013–
Stacy Haiduk
Patty Williams 2009–12, 2015–
Emily Peterson 2009–12
Hunter King
Summer Newman 2012–
Alyvia Alyn Lind
Faith Newman 2011–
Beth Maitland
Traci Abbott 1982–96, 1999, 2001, 2006–
Morgan Obenreder
Crystal 2017–
Dax Randall
Moses Winters 2016–
Marcia Rodd
Myrna Bloodworth 2017–
Tristan Rogers
Colin Atkinson 2010–12, 2014–
Ted Shackelford
William Bardwell 2006–07
Jeffrey Bardwell 2007–
Abhi Sinha
Ravi Shapur 2016–
Max Shippee
Graham Bloodworth 2017–
Terrell Tilford
Barton Shelby 2014–
Jess Walton
Jill Atkinson 1987–
Ray Wise
Ian Ward 2014–
Previous cast members
Main article: List of previous The Young and the Restless cast members
References and notes
^ Jump up to: a b The Young and the Restless Weekdays. "The Young and the Restless Cast: Doug Davidson". Cbs.com. Retrieved September 23, 2012.
Jump up ^ The Young and the Restless Weekdays. "The Young and the Restless Cast: Jeanne Cooper". Cbs.com. Retrieved September 23, 2012.
Jump up ^ The Young and the Restless Weekdays (July 20, 2007). "The Young and the Restless Cast: Eric Braeden". Cbs.com. Retrieved September 23, 2012.
Jump up ^ "Show Guide: Cast Bio: Kate Linder". Sony. Retrieved January 6, 2012.
External links
Soaps.com – Young and Restless
[show]
v
t
e
The Young and the Restless
Cast
Present cast members
Past cast members
Crew
William J. Bell
Lee Phillip Bell
Mal Young
Families
Newman
Abbott
Chancellor
Barber/Winters
Related articles
Genoa City
Storylines
The Bold and the Beautiful
Victor and Nikki
Nick and Sharon
J.T. and Colleen
Daniel and Lily
Devon and Hilary
"Nadia's Theme"
[hide]
v
t
e
The Young and the Restless characters
[show]
Present and future characters
Ashley Abbott
Billy Abbott
Jack Abbott
Jill Abbott
John Abbott
Johnny Abbott
Traci Abbott
Cane Ashby
Charlie Ashby
Mattie Ashby
Sam Ashby
Lily Winters
Colin Atkinson
Fenmore Baldwin
Michael Baldwin
Gloria Abbott Bardwell
Jeffrey Bardwell
Christine Blair
Mariah Copeland
Hilary Curtis
Lauren Fenmore
Kevin Fisher
Scotty Grainger
Devon Hamilton
Reed Hellstrom
Anita Lawson
Dina Mergeron
Chloe Mitchell
Leslie Michaelson
Abby Newman
Chelsea Lawson
Christian Newman
Connor Newman
Faith Newman
Katie Newman
Nicholas Newman
Nikki Newman
Noah Newman
Sharon Newman
Summer Newman
Victor Newman
Victoria Newman
Stitch Rayburn
Barton Shelby
Phyllis Summers
Esther Valentine
Paul Williams
Neil Winters
[show]
Past characters
Kyle Abbott
Kelly Andrews
Genevieve Atkinson
Avery Bailey Clark
Eden Baldwin
Isabella Braña
Mackenzie Browning
Brad Carlton
Colleen Carlton
Daisy Carter
Sheila Carter
Chance Chancellor
Katherine Chancellor
Joe Clark
Matt Clark
Greg Foster
Dr. Snapper Foster
Sam Gibson
Diego Guittierez
Nathan Hastings
Jana Hawkes
J.T. Hellstrom
Juliet Helton
Brittany Hodges
Cole Howard
Diane Jenkins
Mamie Johnson
Leanna Love
Ronan Malloy
Dylan McAvoy
Tucker McCall
Ryan McNeil
Tyler Michaelson
Amber Moore
Adam Newman
Cassie Newman
Cassandra Rawlins
Gina Roma
Daniel Romalotti
Danny Romalotti
Luca Santori
Deacon Sharpe
Marisa Sierras
Rex Sterling
Heather Stevens
Grace Turner
Angelina Veneziano
Ian Ward
Sage Warner
Nina Webster
Patty Williams
Ricky Williams
Hope Wilson
Drucilla Winters
Malcolm Winters
Olivia Winters
[show]
Character lists
1970s
1980s
1990s
2000s
2010
2011
2012
2013
2014
2015
2016
2017
Newman family
Abbott family
Chancellor family
Barber/Winters family
Retrieved from "https://en.wikipedia.org/w/index.php?title=The_Young_and_the_Restless_cast_members&oldid=818651259"
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List of claimed first novels in English
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (January 2008) (Learn how and when to remove this template message)
The following works of literature have each been claimed as the first novel in English.
Thomas Malory, Le Morte d'Arthur, (written c. 1470, published 1485)
William Baldwin, Beware the Cat, (written 1553, published 1570, 1584)[1]
John Lyly, Euphues: The Anatomy of Wit (1578) and Euphues and his England (1580) [2]
Philip Sidney, The Countess of Pembroke's Arcadia (1581)
Margaret Cavendish, The Blazing World (1666)
John Bunyan, The Pilgrim's Progress (1678)[3]
Aphra Behn, Oroonoko (1688)[4]
Daniel Defoe, Robinson Crusoe (1719)[5]
Daniel Defoe, Moll Flanders (1722)
Samuel Richardson, Pamela, or Virtue Rewarded (1740)[6]
The following are some other early long works of prose fiction in English:
William Caxton's 1483 translation of Geoffroy de la Tour Landry, The Book of the Knight of the Tower (originally in French)
Thomas Nashe, The Unfortunate Traveller, or The Life of Jack Wilton (1594)
Jonathan Swift, A Tale of a Tub (1704)
Daniel Defoe, The Consolidator (1705)
Jonathan Swift, Gulliver's Travels (1726)
There are multiple candidates for first novel in English partly because of ignorance of earlier works, but largely because the term novel can be defined so as to exclude earlier candidates:
Some critics require a novel to be wholly original and so exclude retellings like Le Morte d'Arthur.
Most critics distinguish between an anthology of stories with different protagonists, even if joined by common themes and milieus, and the novel (which forms a connected narrative), and so also exclude Le Morte d'Arthur.
Some critics distinguish between the romance (which has fantastic elements) and the novel (which is wholly realistic) and so yet again exclude Le Morte d'Arthur.
Some critics distinguish between the allegory (in which characters and events have political, religious or other meanings) and the novel (in which characters and events stand only for themselves) and so exclude The Pilgrim's Progress and A Tale of a Tub.
Some critics require a novel to have a certain length, and so exclude Oroonoko, defining it instead as a novella.
Some critics distinguish between the picaresque (which has a loosely connected sequence of episodes) and the novel (which has unity of structure) and so exclude The Unfortunate Traveller.
Due to the influence of Ian Watt's seminal study in literary sociology, The Rise of the Novel: Studies in Defoe, Richardson and Fielding (1957), Watt's candidate, Daniel Defoe's Robinson Crusoe (1719), gained wide acceptance.
See also[edit]
The article novel for detailed information about the history of the terms "novel" and "romance" and the bodies of texts they defined in a historical perspective.
Category:Novels by date for earlier claimants in English and other languages.
References[edit]
Jump up ^ Ringler, William A. and Michael Flachmann eds. "Preface." Beware the Cat. San Marino: Huntington Library, 1988.
Jump up ^ Sampson, George (1941). The Concise Cambridge History of English Literature, p. 161. Cambridge University Press. Retrieved 26 April 2014.
Jump up ^ Chapman, J. (1892). The Westminster Review, Volume 138. p. 610.
Jump up ^ Doyle, Laura (2008). Freedom's Empire: Race and the Rise of the Novel in Atlantic Modernity, 1640–1940, p. 97. Duke University Press. Retrieved 26 April 2014.
Jump up ^ Chapman, J. (1892). The Westminster Review, Volume 138. p. 610.
Jump up ^ The New York Times (2007). The New York Times Guide to Essential Knowledge, Second Edition: A Desk Reference for the Curious Mind, p. 67. Macmillan. Retrieved 26 April 2014.
External links[edit]
Historical texts relating to Beware the Cat by William Baldwin
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_claimed_first_novels_in_English&oldid=822172712"
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