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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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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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Early Modern English literature
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Robin Hood: Prince of Thieves
From Wikipedia, the free encyclopedia
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"Prince of Thieves" redirects here. For other uses, see Prince of Thieves (disambiguation).
Robin Hood: Prince of Thieves
Theatrical release poster
Directed by
Kevin Reynolds
Produced by
Pen Densham
Richard Barton Lewis
John Watson
Screenplay by
Pen Densham
John Watson
Story by
Pen Densham
Starring
Kevin Costner
Morgan Freeman
Christian Slater
Alan Rickman
Mary Elizabeth Mastrantonio
Music by
Michael Kamen
Cinematography
Douglas Milsome
Edited by
Peter Boyle
Production
company
Morgan Creek[1]
Distributed by
Warner Bros.
Release date
June 14, 1991 (1991-06-14) (United States)
Running time
143 minutes[2]
Country
United States
Language
English
Budget
$48 million[3]
Box office
$390.5 million[4]
Robin Hood: Prince of Thieves is a 1991 American romantic action adventure film, based on the English folk tale of Robin Hood which originated in the 15th century. The film was directed by Kevin Reynolds. The film's principal cast includes Kevin Costner as Robin Hood, Morgan Freeman as Azeem, Christian Slater as Will Scarlet, Mary Elizabeth Mastrantonio as Maid Marian, and Alan Rickman as the Sheriff of Nottingham.
The film grossed over $390 million worldwide, ranking as the second-highest-grossing film of 1991. For his role as George, Sheriff of Nottingham, Rickman received the BAFTA Award for Best Actor in a Supporting Role. The film's theme song, "(Everything I Do) I Do It for You", by Bryan Adams, was nominated for the Academy Award for Best Song and won the Grammy Award for Best Song Written Specifically for a Motion Picture or Television.[5]
Contents
[hide]
1 Plot
2 Cast
3 Production
3.1 Filming
4 Soundtrack
5 Release
5.1 Reception
5.2 Box office
6 Adaptations
7 See also
8 References
9 External links
Plot[edit]
Robin of Locksley — an English nobleman who joined Richard the Lionheart, King of England in the Third Crusade — is imprisoned in Jerusalem along with his comrade, Peter Dubois. Facing the amputation of Peter's hand by the Ayyubid prison guards, Robin escapes with Peter, saving the life of a Moor named Azeem in the process. Robin, Peter, and Azeem escape through a sewer and into an alley, but Peter is shot and mortally wounded by an archer. Before making his last stand against the approaching guards, Peter has Robin swear to protect his sister, Marian. Robin returns to England with Azeem, who has vowed to accompany him until Azeem's life-debt to Robin is repaid.
In England, with King Richard away in France, the cruel Sheriff of Nottingham rules over the land, aided by his cousin Guy of Gisbourne, the witch Mortianna, and the corrupt Bishop of Hereford. At Locksley Castle, Robin's father, who is loyal to King Richard, is killed by the Sheriff's men after refusing to join them.
Robin returns to England to find his father dead, his home in ruins, and the Sheriff and his men oppressing the people. After telling Marian of Peter's demise, and while fleeing the Sheriff's forces afterwards, Robin and Azeem encounter a band of outlaws hiding in Sherwood Forest, led by Little John. Among the band is Will Scarlet, who holds a belligerent grudge against Robin. Robin assumes command of the group, training them to defend themselves and fight against Nottingham. They rob soldiers and convoys that pass through the forest, then distribute the stolen wealth among the poor. One of their early targets is Friar Tuck, who subsequently joins these Merry Men. Marian begins to sympathize with the band and renders Robin any aid she can muster. Robin's successes infuriate the Sheriff, who increases the mistreatment of the people, resulting in greater local support for Robin Hood.
The Sheriff kills Gisbourne for his failure to prevent the looting of several convoys, and hires Celtic warriors from Scotland to assist his forces in assaulting the hideout. The Sheriff manages to locate the outlaws' hideout and launches an attack, destroying the refuge and capturing most of the outlaws. With the Bishop's help, the Sherrif has Marian confined when she tries to summon help from France. To consolidate his claim to the throne, the Sheriff proposes to Marian (who is Richard's cousin), claiming that if she accepts, he will spare the lives of the captured outlaws. Nevertheless, several of the rebels are due to be executed by hanging as part of the wedding celebration. Among the captured is Will Scarlet, who makes a deal with the Sheriff to find and kill Robin in exchange for his freedom.
Will meets up with Robin and a handful of his aides who survived the Celt's assault. Will informs Robin of the Sheriff's plans to marry Marian and execute Robin's men. Will continues to display anger against Robin, which motivates Robin to question why Will hates him so much. Will then reveals himself to be Robin's younger illegitimate half-brother; Will's mother was a peasant woman with whom Robin's father took comfort after Robin's mother died. Robin's anger toward his father caused him to separate from her and leave Will fatherless. Despite his anger, Robin is overjoyed to learn that he has a brother, and reconciles with Will.
On the day of the wedding and hangings, Robin and his men infiltrate Nottingham Castle, freeing the prisoners. Although Robin's band originally planned to free their friends and retreat, Azeem reveals himself and his willingness to fight the Sheriff, inciting the peasants to revolt. After a fierce fight, Robin kills the Sheriff, but is attacked by Mortianna with a spear. Azeem slays Mortianna, fulfilling his life-debt to Robin. Tuck kills the Bishop by burdening him with treasure and throwing him out a window.
Robin and Marian profess their love for each other and are married in the forest. Their wedding is briefly interrupted by King Richard, who blesses the marriage and thanks Robin for his deeds.
Cast[edit]
Kevin Costner as Robin Hood (Robin of Locksley)[6]
Morgan Freeman as Azeem
Christian Slater as Will Scarlet
Alan Rickman as George, Sheriff of Nottingham[7]
Mary Elizabeth Mastrantonio as Marian Dubois
Geraldine McEwan as Mortianna
Mike McShane as Friar Tuck
Brian Blessed as Lord Locksley
Michael Wincott as Guy of Gisborne
Nick Brimble as Little John
Harold Innocent as the Bishop of Hereford
Walter Sparrow as Duncan
Daniel Newman as Wulf
Sean Connery (uncredited) as King Richard
Jack Wild as Much the miller's son
Daniel Peacock as Bull
Production[edit]
Sycamore Gap at Hadrian's Wall, today known as the "Robin Hood Tree"[8]
Rickman turned down the role of the sheriff twice before he was told he could have carte blanche with his interpretation of the character.
On the DVD commentary for ITV's Robin of Sherwood television series, which ran from 1983 to 1986, writer and creator Richard Carpenter explains that the stunt coordinator from the Robin of Sherwood series, Terry Walsh, was hired to do stunt work for Prince of Thieves. While on set, Walsh noticed a Saracen assassin character named Nasir was in the film. That character was a creation of Carpenter and is exclusive to the Robin of Sherwood series. Once the creators of Prince of Thieves realized a potential for copyright infringement, they changed the character's name from Nasir to Azeem. Carpenter also explains that Costner and others involved in Prince of Thieves have admitted to watching Robin of Sherwood as inspiration for their film.[citation needed]
Filming[edit]
Principal exteriors were shot on location in the United Kingdom. A second unit filmed the medieval walls and towers of the Cité de Carcassonne in the town of Carcassonne in Aude, France, for the portrayal of Nottingham and its castle. Locksley Castle was Wardour Castle in Wiltshire – restored in an early shot using a matte painting. Marian's manor was filmed at Hulne Priory in Northumberland. Scenes set in Sherwood Forest were filmed throughout England: Burnham Beeches in Buckinghamshire was used for the outlaws' encampment, Aysgarth Falls in Yorkshire for the fight scene between Robin and Little John, and Hardraw Force in North Yorkshire was the location where Marian sees Robin bathing.[9] Sycamore Gap on Hadrian's Wall was used for the scene when Robin first confronts the sheriff's men.[10] Chalk cliffs at Seven Sisters, Sussex were used as the locale for Robin's return to England from the Crusades.[11]
Interior scenes were completed at Shepperton Studios in Surrey.[9]
Soundtrack[edit]
The original music score was composed, orchestrated and conducted by Michael Kamen. An excerpt from the main title music was subsequently used as the logo music for Morgan Creek,[12] and has been used by Walt Disney Home Video in intros for trailers on DVD/Blu-ray.[13]
Robin Hood: Prince of Thieves (Original Soundtrack)
Film score by Michael Kamen
Released
July 2, 1991
Length
60:22
Label
Morgan Creek Productions
"Overture/A Prisoner of the Crusades" (8:27)
"Sir Guy of Gisborne/The Escape to Sherwood" (7:27)
"Little John/The Band in the Forest" (4:52)
"The Sheriff and His Witch" (6:03)
"Maid Marian" (2:57)
"Training/Robin Hood, Prince of Thieves" (5:15)
"Marian at the Waterfall" (5:34)
"The Abduction/The Final Battle at the Gallows" (9:53)
"(Everything I Do) I Do It for You" – Bryan Adams (6:33)
"Wild Times" – Jeff Lynne (3:12)
Release[edit]
Robin Hood: Prince of Thieves was submitted for classification from the British Board of Film Classification and required 14 seconds to be cut from the film to obtain a PG rating.[2]
Reception[edit]
Chicago Sun-Times critic Roger Ebert praised Freeman's performance as well as Rickman's, but ultimately decried the film as a whole, giving it two stars and stating, "Robin Hood: Prince of Thieves is a murky, unfocused, violent, and depressing version of the classic story... The most depressing thing about the movie is that children will attend it expecting to have a good time."[14] The New York Times gave the film a negative review, with Vincent Canby writing that the movie is "a mess, a big, long, joyless reconstruction of the Robin Hood legend that comes out firmly for civil rights, feminism, religious freedom, and economic opportunity for all."[15] The Los Angeles Times found the movie unsatisfactory, as well.[16] Costner was criticised for not attempting an English accent.[17]
Lanre Bakare, writing in The Guardian, calls Rickman's Sheriff, for which he won a BAFTA, a "genuinely great performance".[18] Review aggregation website Rotten Tomatoes retrospectively collected reviews from 53 critics to give the film a score of 49%.[19]
Prince of Thieves was nominated for two Golden Raspberry Awards: Kevin Costner won the Worst Actor award for his performance as Robin Hood, while Christian Slater received a nomination for Worst Supporting Actor for his performances in this film and Mobsters, but lost to Dan Aykroyd for Nothing but Trouble.[20]
In 2005, the American Film Institute nominated this film for AFI's 100 Years of Film Scores.[21]
Box office[edit]
The film grossed $25 million in its opening weekend and $18.3 million in its second. The film eventually made $390,493,908 at the global box office, making it the second-highest grossing film of 1991, immediately behind Terminator 2: Judgment Day. It enjoyed the second-best opening for a nonsequel, at the time.[22][23][24][25]
Adaptations[edit]
A tie-in video game of the same name was released in 1991 for the Nintendo Entertainment System and Game Boy. Developed by Sculptured Software Inc. and Bits Studios, respectively, and published by Virgin Games, Inc., it was featured as the cover game for the July 1991 issue of Nintendo Power magazine.[26]
A toy line was released by Kenner, consisting of action figures and playsets. Notably, all but one of the figures were slightly modified from Kenner's well known Super Powers line, while Friar Tuck, as well as the vehicles and playset, were modified from Star Wars: Return of the Jedi toys.[27]
See also[edit]
Film in the United States portal
1990s portal
Princess of Thieves – 2001 television movie
Robin Hood – English folk tale
Robin Hood: Men in Tights – 1993 parody film
Robin Hood – 1991 British film
References[edit]
Jump up ^ Easton, Nina J. (1990-07-24). "Costner May Put Morgan Creek Ahead of Robin Hood Pack". The Los Angeles Times. Retrieved 2010-10-02.
^ Jump up to: a b "ROBIN HOOD - PRINCE OF THIEVES (PG) (CUT)". British Board of Film Classification. July 4, 1991. Retrieved January 19, 2016.
Jump up ^ Billington, Michael (March 18, 1991). "Robin Hood Freshens Up A Film Legend". Orlando Sentinel. Retrieved March 22, 2017.
Jump up ^ "Robin Hood: Prince of Thieves (1991)". Box Office Mojo. 1991-10-17. Retrieved 2016-10-29.
Jump up ^ "1992 Grammy Awards". metrolyrics.com. Archived from the original on February 9, 2009. Retrieved January 1, 2009.
Jump up ^ Dowd, Maureen (1991-06-09). "FILM; Hollywood's Superhunk Heads for Nottingham". The New York Times. Retrieved 2010-10-02.
Jump up ^ Leydon, Joe (1991-06-09). "Robin Hood' and the uncertain science of hype". The Los Angeles Times. Retrieved 2010-10-02.
Jump up ^ Sycamore Gap, a section of the wall between two crests just east of Milecastle 39, is locally known as the "Robin Hood Tree" for its use in the 1991 film Robin Hood: Prince of Thieves (1991).
^ Jump up to: a b Pearce, Garth; Green, Simon (1991). Robin Hood: Prince of Thieves. Bdd Promotional Book Co. pp. 22–34. ISBN 9780792456339.
Jump up ^ Else, David & Sandra Bardwell, Belinda Dixon, Peter Dragicevich (2007). Lonely Planet: Walking in Britain. Lonely Planet. p. 224. ISBN 978-1-7410-4202-3.
Jump up ^ Pirani, Adam (May 1991). "Robin Hood: Prince of Thieves". Starlog. p. 40.
Jump up ^ David Victor (August 30, 2012). "Studio Logo Music". Retrieved November 25, 2015.
Jump up ^ "Film Score Monthly". July 10, 2009. Retrieved November 25, 2017.
Jump up ^ "Robin Hood: Prince of Thieves". Chicago Sun Times. Retrieved 2010-10-02.
Jump up ^ Canby, Vincent (1991-06-14). "A Polite Robin Hood In a Legend Recast". The New York Times. Retrieved 2013-02-21.
Jump up ^ Turan, Kenneth (1991-06-14). "'Robin': Medieval Dash, New Age Muddle". The Los Angeles Times. Retrieved 2010-10-02.
Jump up ^ Easton, Nina J. (1991-06-23). "A look inside Hollywood and the movies". The Los Angeles Times. Retrieved 2010-10-02.
Jump up ^ Bakare, Lanre. "My guilty pleasure – Robin Hood: Prince of Thieves", The Guardian, March 26, 2014
Jump up ^ "Robin Hood: Prince of Thieves (1991)". Rotten Tomatoes. Retrieved 2013-06-10.
Jump up ^ Wilson, John (2005). The Official Razzie Movie Guide: Enjoying the Best of Hollywood's Worst. Grand Central Publishing. ISBN 0-446-69334-0.
Jump up ^ "AFI's 100 Years of Film Scores Nominees" (PDF). Retrieved 2016-08-07.
Jump up ^ "Robin Hood prince of summer flicks with $18.3 million weekend". Baltimore Sun. Retrieved 2010-10-02.
Jump up ^ Fox, David J. (1991-06-25). "Robin Hood Still Riding Ahead of Box Office Pack". The Los Angeles Times. Retrieved 2010-10-02.
Jump up ^ Fox, David J. (1991-06-18). "'Robin' Hits Impressive Box Office Bull's-Eye". The Los Angeles Times. Retrieved 2010-10-02.
Jump up ^ "Can 'Robin Hood' Keep Up Its Box-office Momentum?". Orlando Sentinel. Retrieved 2010-10-02.
Jump up ^ Tilden, Gail, ed. (July 1991). Nintendo Power. Vol 26. ISSN 1041-9551.
Jump up ^ Salvatore, Ron. "The recycling of the Force - Starwars". The Star Wars Collectors Archive. Retrieved 2016-02-06.
External links[edit]
Wikiquote has quotations related to: Robin Hood: Prince of Thieves
Robin Hood: Prince of Thieves on IMDb
Robin Hood: Prince of Thieves at the TCM Movie Database
Robin Hood: Prince of Thieves at Box Office Mojo
Robin Hood: Prince of Thieves at Rotten Tomatoes
"The Battle of Sherwood Forest", a 1991 Entertainment Weekly cover story about the film's tumultuous production.
Robin Hood: Prince of Thieves Transcript
[show]
v
t
e
Robin Hood
Characters
Robin Hood
Maid Marian
Merry Men
Much the Miller's Son
Little John
Friar Tuck
Alan-a-Dale
Will Scarlet
Will Stutely
Gilbert Whitehand
Arthur a Bland
David of Doncaster
The Jolly Pinder of Wakefield
Sheriff of Nottingham
Guy of Gisbourne
Prince John
Bishop of Hereford
Richard at the Lee
King Richard
Settings
Sherwood Forest
Nottingham
Loxley
Barnsdale
Wentbridge
Screen
Film
Robin Hood (1912)
Robin Hood (1922)
The Adventures of Robin Hood (1938)
The Bandit of Sherwood Forest (1946)
The Prince of Thieves (1948)
The Story of Robin Hood and His Merrie Men (1952)
The Men of Sherwood Forest (1954)
Sword of Sherwood Forest (1960)
A Challenge for Robin Hood (1967)
The Scalawag Bunch (1971)
Wolfshead: The Legend of Robin Hood (1973)
The Arrows of Robin Hood (1975)
Robin and Marian (1976)
Aaj Ka Robin Hood (1988)
O Mistério de Robin Hood (1990)
Robin Hood (1991)
Robin Hood: Prince of Thieves (1991)
Robin Hood (2010)
Robin Hood (2018)
TV
Robin Hood (1953)
The Adventures of Robin Hood (1955)
The Legend of Robin Hood (1968)
The Legend of Robin Hood (1975)
Robin of Sherwood (1984)
The New Adventures of Robin Hood (1997)
Robin Hood (2006)
Animated
Robin Hood Makes Good (1939)
Rabbit Hood (1949)
Robin Hood Daffy (1958)
Robin Hoodwinked (1958)
Robin Hood (1973)
Robin Hood (1990)
Young Robin Hood (1991)
Tom and Jerry: Robin Hood and His Merry Mouse (2012)
Parody
When Things Were Rotten (1975)
The Zany Adventures of Robin Hood (1984)
Maid Marian and Her Merry Men (1989)
Robin Hood: Men in Tights (1993)
Alternate
settings
Mexicali Rose (1939 film)
Robin and the 7 Hoods (1964 film)
Naan Sigappu Manithan (1985 Tamil film)
Nyayam Meere Cheppali (1985 Telugu film)
Catch Me Now (2008 Chinese TV series)
Alyas Robin Hood (2016 Philippines TV Series)
Popular culture
Robin Hood (Once Upon a Time character)
Robin Hood (DC Comics character)
Child ballads
8: Erlinton
102: Willie and Earl Richard's Daughter
103: Rose the Red and White Lily
115: Robyn and Gandeleyn
117: A Gest of Robyn Hode
118: Robin Hood and Guy of Gisborne
119: Robin Hood and the Monk
120: Robin Hood's Death
121: Robin Hood and the Potter
123: Robin Hood and the Curtal Friar
124: The Jolly Pinder of Wakefield
126: Robin Hood and the Tanner
127: Robin Hood and the Tinker
128: Robin Hood Newly Revived
129: Robin Hood and the Prince of Aragon
130: Robin Hood and the Scotchman
131: Robin Hood and the Ranger
132: The Bold Pedlar and Robin Hood
136: Robin Hood's Delight
138: Robin Hood and Allan-a-Dale
139: Robin Hood's Progress to Nottingham
140: Robin Hood Rescuing Three Squires
141: Robin Hood Rescuing Will Stutly
142: Little John a Begging
143: Robin Hood and the Bishop
144: Robin Hood and the Bishop of Hereford
145: Robin Hood and Queen Katherine
146: Robin Hood's Chase
147: Robin Hood's Golden Prize
148: The Noble Fisherman
149: The Noble Fisherman
151: The King's Disguise, and Friendship with Robin Hood
152: Robin Hood and the Golden Arrow
153: Robin Hood and the Valiant Knight
154: A True Tale of Robin Hood
Stage / Theatre
The Downfall and The Death of Robert Earl of Huntington (1598 and 1601 plays)
The Merrie Men of Sherwood Forest (1871 operetta)
Robin Hood (1890 opera)
The Foresters (1892 play)
Twang!! (1965 musical parody)
Robin Hood (1934 opera)
Robin Hood (1998 ballet)
Robin des Bois (2013 musical)
Video games
Robin of the Wood (1985)
The Curse of Sherwood (1987)
The Adventures of Robin Hood (1991)
Conquests of the Longbow: The Legend of Robin Hood (1991)
Robin Hood: The Legend of Sherwood (2002)
Robin Hood: Defender of the Crown (2003)
Volume (2015)
Literature
Ivanhoe (1819)
Maid Marian (1822)
The Merry Adventures of Robin Hood (1883)
Bows against the Barons (1934)
The Once and Future King (1958)
The Outlaws of Sherwood (1988)
Through a Dark Mist (1991)
Lady of the Forest (1992)
In the Shadow of Midnight (1994)
The Last Arrow (1997)
Lady of Sherwood (1999)
Ronin Hood of the 47 Samurai (2005)
King Raven Trilogy (2006)
Music
Legend (1984 soundtrack)
Robin Hood (2006 soundtrack)
Robin Hood – czwarta strzała (1997)
"Love" (song)
"Not in Nottingham" (song)
"(Everything I Do) I Do It for You" (song)
The Tale of Gamelyn
Alan Dale
Outlaw (2009)
Holy Warrior (2010)
King's Man (2011)
Warlord (2012)
Grail Knight (2013)
The Iron Castle (2014)
The King's Assassin (2015)
The Death of Robin Hood (2016)
Related
Miss Robin Hood
Son of the Guardsman
The Son of Robin Hood
The Bandit of Sherwood Forest
Princess of Thieves
Robin Hood Morality Test
"Robot of Sherwood"
"Robin Good and His Not-So-Merry Men"
Once Upon a Time
[show]
v
t
e
Films directed by Kevin Reynolds
Fandango (1985)
The Beast (1988)
Robin Hood: Prince of Thieves (1991)
Rapa-Nui (1994)
Waterworld (1995)
One Eight Seven (1997)
The Count of Monte Cristo (2002)
Tristan & Isolde (2006)
Hatfields & McCoys (2012)
Risen (2016)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Robin_Hood:_Prince_of_Thieves&oldid=830862296"
Categories:
1991 films
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1991 soundtracks
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American action films
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American films
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Films scored by Michael Kamen
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Michael Kamen albums
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West Area Computers
From Wikipedia, the free encyclopedia
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West Area Computing Unit
Nationality
American
Other names
West Area Computers
Occupation
NASA Mathematicians
Years active
1943–1958
The West Area Computing Unit (West Area Computers) was an all-African American all-female group of mathematicians who worked as human computers at the Langley Research Center of NACA (predecessor of NASA) from 1943 through 1958. The group, a subset of the hundreds of female mathematicians who began careers in aeronautical research during World War II, was originally subject to Virginia's Jim Crow laws, requiring them to use segregated bathrooms and cafeterias.[1]
The Langley Research Center of NACA began recruiting African-American women with college degrees, to work as computers, during the 1940s. The West Area Computers's responsibilities included processing data, as well as joining other teams at NACA on a temporary basis as needed.[1]
According to an unpublished study by Beverly E. Golemba of Langley's early computers, a number of other women did not know about the West Computers.[2] That said, both the black and white women Golemba interviewed recalled that when computers from both groups were assigned to a project together, “everyone worked well together.”[3][1][4]
Originally supervised by white women, in 1949 Dorothy Vaughan was put in charge, becoming NASA's first African-American manager. Vaughan was a mathematician who worked at Langley from 1943 through her retirement in 1971.[5]
In 1958, when the NACA made the transition to NASA, segregated facilities, including the West Computing office, were abolished.[5]
Notable members[edit]
Mathematician Katherine Johnson, who in 2015 was named a Presidential Medal of Freedom recipient, joined the West Area Computing group in 1953. She was subsequently reassigned to Langley's Flight Research Division, where she performed notable work including providing the trajectory analysis for astronaut John Glenn's MA-6 Project Mercury orbital spaceflight.[6] Mary Jackson also worked in the West Area Computing Unit,[7] and the work of all three women (Vaughan, Johnson, and Jackson) is featured in the 2016 film Hidden Figures.
See also[edit]
Melba Roy Mouton
References[edit]
^ Jump up to: a b c McLennan, Sarah (2011). "Human Computers". NASA Cultural Resources. Retrieved 6 February 2017.
Jump up ^ "Beverly E. Golemba Papers, MS 307, Archives of Women in Science and Engineering, Special Collections Department". Iowa State University Library. Retrieved 6 February 2017.
Jump up ^ ""Panel Discussion with Women Computers" NASA Langley videotape". December 13, 1990.
Jump up ^ Beverly Golemba, Human Computers: The Women in Aeronautical Research, unpublished manuscript 1994, NASA Langley Archives
^ Jump up to: a b Loff, Sarah (22 November 2016). "Dorothy Vaughan Biography". NASA. Retrieved 6 February 2017.
Jump up ^ "West Area Computers - Trailblazers in American Space History" (PDF). Center for the History of Physics at AIP. Summer 2014. Retrieved 6 February 2017.
Jump up ^ Shetterly, Margot Lee (2016-09-06). Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race. William Morrow. ISBN 9780062363596.
Retrieved from "https://en.wikipedia.org/w/index.php?title=West_Area_Computers&oldid=826619533"
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