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In linear algebra, a linear function is a map f between two vector spaces such that. Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself. In other terms the linear function preserves vector addition and scalar multiplication.
A linear function is a polynomial function in which the variable x has degree at most one: [2] . Such a function is called linear because its graph, the set of all points in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below). If the slope is , this is a constant function defining a ...
Differentiation is linear. For any functions and and any real numbers and , the derivative of the function with respect to is: In Leibniz's notation this is written as: Special cases include: The constant factor rule. The sum rule.
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 section. If b = 0, the line is a vertical line (that is a line parallel to ...
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, 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 ...
The simplest way for writing the chain rule in the general case is to use the total derivative, which is a linear transformation that captures all directional derivatives in a single formula. Consider differentiable functions f : R m → R k and g : R n → R m, and a point a in R n.
Order of operations. In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operations. The rank of an operation is called its precedence, and ...
The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).