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Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
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 ...
The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any ordinary point of a homogeneous linear differential equation of order n {\displaystyle n} there exists a fundamental system of n {\displaystyle n ...
vector spaces, whose study is essentially the same as linear algebra; ring theory; commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry; homological algebra; Lie algebra and Lie group theory
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.
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...
The number of brackets is the number of linear equations associated with an integral. This term reflects the common practice of bracketing each linear equation. [15] The complexity index is the number of integrand sums minus the number of brackets (linear equations). Each series expansion of the integrand contributes one sum.