lineaarimuutos

Lineaarimuutos, also known as a linear transformation, is a fundamental concept in linear algebra. It is 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 linear structure of these spaces.

To be considered a linear transformation, a function T must satisfy two key properties for any vectors u and v in the domain space and any scalar c:

1. T(u + v) = T(u) + T(v) (Additivity)

2. T(c * u) = c * T(u) (Homogeneity)

These properties ensure that the transformation does not "break" the linearity of the vector spaces. Linear transformations are ubiquitous in mathematics, physics, computer graphics, and many other scientific fields. They are often represented by matrices, where multiplying a vector by a matrix performs the linear transformation. Common examples include rotations, reflections, scaling, and shearing of geometric objects. Understanding linear transformations is crucial for solving systems of linear equations, analyzing data, and developing algorithms in various computational disciplines.