Lineaarteisendus

Lineaarteisendus is a mathematical concept that describes a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In simpler terms, a linear transformation maps vectors from one space to another in a way that maintains the structure of these spaces. Specifically, for any two vectors u and v in the first vector space and any scalar c, a function T is a linear transformation if T(u + v) = T(u) + T(v) and T(c * u) = c * T(u).

Linear transformations are fundamental in linear algebra and have wide-ranging applications in various fields. They can be represented by matrices, and the properties of the transformation are directly related to the properties of its corresponding matrix. For example, the dimension of the image of a linear transformation is related to the rank of its matrix.

Common examples of linear transformations include rotations, reflections, scaling, and shearing in geometric spaces. They are used extensively in computer graphics for manipulating objects, in physics for describing physical phenomena like wave propagation, and in engineering for solving systems of equations. Understanding linear transformations is crucial for comprehending many advanced mathematical and scientific concepts.