Multilineárnom

Multilineárnom is a Slovak term that translates to "multilinear" in English. It refers to a function or mapping that is linear in each of its arguments independently. This means that if you fix all but one argument, the function behaves as a linear transformation with respect to that single varying argument.

For instance, consider a function f(x, y) that takes two arguments. If f is multilinear, then for any fixed value of y, the function g(x) = f(x, y) is a linear function of x. Similarly, for any fixed value of x, the function h(y) = f(x, y) is a linear function of y. This property must hold for all possible fixed values of the other arguments.

Multilinearity is a fundamental concept in various areas of mathematics, including linear algebra, differential geometry, and abstract algebra. In linear algebra, multilinear forms are a key example. A multilinear form is a function that takes multiple vectors as input and returns a scalar, and it is linear in each vector argument. Determinants are a classic example of a multilinear form.

The concept can be extended to functions with any number of arguments. A function f(x1, x2, ..., xn) is multilinear if it is linear in x1 when x2, ..., xn are held constant, linear in x2 when x1, x3, ..., xn are held constant, and so on for all arguments. This property is crucial for understanding and manipulating complex mathematical structures where the interaction between multiple variables needs to be analyzed in a structured, linear way.