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Independence system. In combinatorial mathematics, an independence system is a pair , where is a finite set and is a collection of subsets of (called the independent sets or feasible sets) with the following properties: The empty set is independent, i.e., . (Alternatively, at least one subset of is independent, i.e., .)
Linear independence. In the theory of vector spaces, a set of vectors is said to be linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be linearly dependent. These concepts are central to the definition of dimension.
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 ...
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent[ 1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect ...
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. [ 1] The concept typically arises in the context of linear equations. If it is possible to duplicate one of the equations in a system by multiplying each of the other equations by some number (potentially ...
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, a function is a rule for taking an input (in the simplest case, a number or set of numbers) [5] and providing an output (which may also be a number). [5] A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. [6]
Basis (linear algebra) The same vector can be represented in two different bases (purple and red arrows). In mathematics, a set B of vectors in a vector space V is called a basis ( pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are ...