A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the original vector space. It is also known as a linear map or linear operator.
A linear transformation can be represented by a matrix that contains the coefficients of the transformation. It can also be represented by a set of equations that describe how each element of the input vector is transformed.
Some common properties of a linear transformation are: it preserves the origin, it preserves addition and scalar multiplication, and it preserves the structure of the vector space (e.g. lines remain lines and planes remain planes).
To apply a linear transformation to a vector, you multiply the vector by the transformation matrix or plug the vector into the transformation equations. The resulting vector will be the transformed vector in the new vector space.
Linear transformations are used in many fields, including physics, engineering, computer graphics, and economics. Some examples of real-life applications include image and signal processing, geometric transformations in 3D computer graphics, and solving systems of linear equations in economics and business.