Introduction
In linear algebra, a linear transformation (also called a linear map, linear operator, or vector space homomorphism) is a function between two vector spaces that preserves the vector space structure. In simpler terms, it's a rule that maps vectors from one space to another, while respecting the operations of addition and scalar multiplication in those spaces.
Here's a more formal definition:
Let V and W be vector spaces over the same field (usually the real numbers or complex numbers). A linear transformation T is a function from V to W such that for any vectors v₁ and v₂ in V and any scalar a in the field:
- Preserves addition: T(v₁ + v₂) = T(v₁) + T(v₂)
- Preserves scalar multiplication: T(av) = aT(v)
These properties capture the essence of linearity: scaling a vector by a scalar corresponds to scaling its image under the transformation, and adding vectors corresponds to adding their images.
Linear transformations are fundamental concepts in linear algebra, with numerous applications in various fields, including physics, engineering, computer graphics, and economics. They provide a powerful framework for analyzing and manipulating systems that exhibit linear relationships.
Here are some key aspects of linear transformations:
- Matrix Representation: Linear transformations can be represented by matrices. The specific matrix depends on the chosen bases for the vector spaces involved.
- Kernel and Image: The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector. The image of a transformation is the set of all vectors in the codomain that are the output of the transformation for some vector in the domain.
- Examples: Scaling, rotations, and projections are all examples of linear transformations.