Transformationsmatriserna

Transformationsmatriserna, or transformation matrices in English, are fundamental tools in linear algebra used to represent linear transformations between vector spaces. A linear transformation is a function that maps vectors from one vector space to another while preserving vector addition and scalar multiplication. A transformation matrix encapsulates this mapping in a compact and systematic way.

For a transformation from an n-dimensional vector space to an m-dimensional vector space, the transformation matrix will have dimensions m x n. If we have a vector v in the n-dimensional space, and T is the linear transformation, then the transformed vector Tv in the m-dimensional space can be obtained by multiplying the transformation matrix A by the vector v: Tv = Av. The columns of the transformation matrix A correspond to the images of the standard basis vectors of the input space.

These matrices are crucial for various applications. In computer graphics, they are used to perform operations such as translation, rotation, scaling, and shearing of objects in 2D and 3D space. In physics and engineering, they model physical processes, such as changes in state or the behavior of systems under certain conditions. Understanding transformation matrices allows for efficient computation and analysis of linear transformations in a wide range of scientific and technical fields.