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This notion has made it possible to use the methods of graph theory in universal algebra and several other areas of discrete mathematics and computer science.Graph algebras have been used, for example, in constructions concerning dualities, [2] equational theories, [3] flatness, [4] groupoid rings, [5] topologies, [6] varieties, [7] finite-state machines, [8] [9] tree languages and tree ...
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). 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]
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
If, in addition to vector addition and scalar multiplication, there is a bilinear vector product V × V → V, the vector space is called an algebra; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).
It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers ...
Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. [1] The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.