Lineaarikuvaukset

Lineaarikuvaukset, also known as linear transformations, are fundamental concepts in linear algebra. A linear mapping or linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. Specifically, if $T$ is a linear transformation from vector space $V$ to vector space $W$, then for any vectors $u$ and $v$ in $V$ and any scalar $c$, the following two properties must hold: $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$.

These properties ensure that the structure of the vector spaces is maintained under the transformation. Linear transformations are the "straight" mappings between vector spaces; they do not bend or curve the space. Geometrically, they can be represented by operations such as rotations, reflections, scaling, and shearing. In finite-dimensional vector spaces, linear transformations can be completely described by matrices. Multiplying a vector by a matrix represents the application of the corresponding linear transformation.

The set of all linear transformations from $V$ to $W$ forms a vector space itself. Important linear transformations include the identity transformation, which maps every vector to itself, and the zero transformation, which maps every vector to the zero vector. Understanding linear transformations is crucial for solving systems of linear equations, analyzing data, and many other applications in mathematics, physics, engineering, and computer science.