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Some authors use "linear function" only for linear maps that take values in the scalar field; [6] these are more commonly called linear forms. The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0, ..., 0) = 0, or, equivalently, when the constant b equals zero in the one-degree polynomial above. Geometrically, the ...
There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality. In geometric terms, the feasible region defined by all values of x {\displaystyle \mathbf {x} } such that A x ≤ b {\textstyle A\mathbf {x} \leq \mathbf {b} } and ∀ i , x i ≥ 0 ...
Every linear programming problem, referred to as a primal problem, can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. In matrix form, we can express the primal problem as: Maximize c T x subject to Ax ≤ b, x ≥ 0; with the corresponding symmetric dual problem,
A linear form is a linear map from a vector space V over a field F to the field of scalars F, viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V, and usually denoted V* [17] or V ′. [18] [19]
Conversely, every line is the set of all solutions of a linear equation. 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 ...
A module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions. A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group.
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