Lösungsvektor

Lösungsvektor is a term originating from German that translates to "solution vector" in English. It is most commonly used in the context of mathematics and computer science, particularly in the realm of solving systems of equations or optimization problems.

In linear algebra, a solution vector represents a set of values that, when substituted into the variables of a system of linear equations, satisfies all equations simultaneously. For example, if you have a system like 2x + y = 5 and x - y = 1, the solution vector would be [2, 1], as x=2 and y=1 makes both equations true. The existence and uniqueness of a solution vector depend on the properties of the system, such as the rank of the coefficient matrix.

In optimization, a solution vector refers to a set of decision variables that yields the optimal value (either maximum or minimum) of an objective function, subject to a given set of constraints. For instance, in a linear programming problem, the solution vector would be the specific quantities of different products to manufacture to maximize profit while adhering to resource limitations. Finding this vector often involves algorithms like the simplex method or gradient descent.

The concept of a solution vector is fundamental to understanding the results of various mathematical models and computational processes, providing a clear representation of the outcome that fulfills the defined conditions.