A linear transformation is a mathematical operation that maps a set of points or vectors from one space to another, while preserving certain properties such as linearity and proportionality.
The key characteristics of a linear transformation include preserving the origin, preserving lines, and preserving proportions. This means that the transformation does not change the location of the origin, straight lines remain straight after transformation, and the ratio of distances between points is maintained.
A linear transformation can be represented using a matrix or a system of equations. In matrix form, the transformation is represented as a square matrix that operates on a vector to produce a new vector. In equation form, the transformation is represented as a set of linear equations that relate the input variables to the output variables.
The main difference between a linear transformation and a non-linear transformation is that a linear transformation preserves linearity and proportionality, while a non-linear transformation does not. This means that a non-linear transformation can change the shape of objects or distort their proportions, whereas a linear transformation will not.
Linear transformations have many real-life applications, including in computer graphics, data compression, and engineering. For example, in computer graphics, linear transformations are used to rotate, scale, and translate 2D and 3D objects. In data compression, linear transformations are used to reduce the amount of data needed to represent an image or signal. In engineering, linear transformations are used to model physical systems and analyze their behavior.