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Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...
An example of a linear function is the function defined by () = (,) that maps the real line to a line in the Euclidean plane R 2 that passes through the origin. An example of a linear polynomial in the variables X , {\displaystyle X,} Y {\displaystyle Y} and Z {\displaystyle Z} is a X + b Y + c Z + d . {\displaystyle aX+bY+cZ+d.}
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
Linear functions commonly arise from practical problems involving variables , with a linear relationship, that is, obeying a linear equation + =. If B ≠ 0 {\displaystyle B\neq 0} , one can solve this equation for y , obtaining
With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including geometric transformations, coordinate changes, quadratic forms, and many other part of mathematics.
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. [7] 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
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 ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.