A linear transformation is a mathematical function that maps a vector space to another vector space while preserving the basic structure of the original space. In other words, it is a transformation that preserves lines and the origin in a vector space.
Linear transformations have two main properties: additivity and homogeneity. Additivity means that the transformation of the sum of two vectors is equal to the sum of the transformed vectors, and homogeneity means that the transformation of a vector multiplied by a scalar is equal to the scalar multiplied by the transformed vector.
A linear transformation can be represented by a matrix. The transformation of a vector is obtained by multiplying the matrix with the vector. The matrix is often referred to as the transformation matrix or the standard matrix of the linear transformation.
The main difference between linear and non-linear transformations is that linear transformations preserve the basic structure of the vector space, while non-linear transformations do not. This means that non-linear transformations do not preserve lines and the origin in a vector space.
Linear transformations have many applications in mathematics, physics, and engineering. They are used to solve systems of linear equations, find eigenvalues and eigenvectors, and in data analysis and image processing. They are also used in computer graphics to manipulate 3D objects and in machine learning for feature extraction and dimensionality reduction.