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In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
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.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
The problem can be solved by reduction to the minimum cost network flow problem. [11] Construct a flow network with the following layers: Layer 1: One source-node s. Layer 2: a node for each agent. There is an arc from s to each agent i, with cost 0 and capacity c i. Level 3: a node for each task.
When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...
These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathematical analysis , the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product , norm , or topology ) and the ...
In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes . The natural domain of a linear function f ( x ) {\displaystyle f(x)} , the set of allowed input values for x , is the entire set of real numbers , x ∈ R ...
The hidden linear function problem, is a search problem that generalizes the Bernstein–Vazirani problem. [1] In the Bernstein–Vazirani problem, the hidden function is implicitly specified in an oracle; while in the 2D hidden linear function problem (2D HLF), the hidden function is explicitly specified by a matrix and a binary vector. 2D HLF can be solved exactly by a constant-depth quantum ...