Lineaarikuvaus

Lineaarikuvaus is a Finnish term that translates to linear transformation or linear map in English. It is a fundamental concept in linear algebra and 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 the vector spaces.

Formally, a function L from a vector space V to a vector space W is called a linear transformation if for any vectors u and v in V and any scalar c, the following two conditions hold: L(u + v) = L(u) + L(v) and L(c * u) = c * L(u). These properties ensure that the "linearity" of the vector spaces is preserved under the transformation. Examples of linear transformations include rotations, scaling, and reflections in geometric spaces.

Linear transformations can be represented by matrices. If V and W are finite-dimensional vector spaces, and a basis is chosen for each, then any linear transformation between them can be uniquely represented by a matrix. Multiplying a vector by this matrix performs the linear transformation. The study of linear transformations and their properties is central to many areas of mathematics, physics, engineering, and computer science. They are essential for understanding systems of linear equations, solving differential equations, and analyzing data.