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A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions.
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The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. [5]
The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C. [ 4 ] In modern terms, it is a subsystem of the linear system associated to the normal bundle to C ↪ Y {\displaystyle C\hookrightarrow Y} .
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a concatenation of column vectors. For example, when solving the linear system =, rather than understanding x as the product of with b, it is helpful to think of x as the vector of coefficients in the linear expansion of b in the basis formed by the columns of A.
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A computer network diagram is a schematic depicting the nodes and connections amongst nodes in a computer network or, more generally, any telecommunications network. Computer network diagrams form an important part of network documentation.