A linear transformation is a mathematical function that maps a set of input vectors to a set of output vectors in a linear manner. This means that the output vector is a scaled and/or rotated version of the input vector, and the transformation follows the rules of linear algebra.
In mathematics, a linear transformation is typically represented by a matrix. Each column of the matrix represents the transformation of a basis vector in the input space to a basis vector in the output space. The matrix can also be used to perform the transformation on any vector in the input space.
A linear transformation must follow two main properties: additivity and homogeneity. Additivity means that the transformation of a sum of two vectors is equal to the sum of their individual transformations. Homogeneity means that the transformation of a scaled vector is equal to the scaled transformation of the original vector.
As mentioned before, a linear transformation is represented by a matrix. This means that any linear transformation can also be represented as a matrix multiplication. The matrix that represents the transformation is called the transformation matrix, and it can be used to perform the transformation on any vector in the input space.
Linear transformations have many real-world applications, such as in computer graphics, data analysis, and physics. In computer graphics, linear transformations are used to rotate, scale, and translate objects in a 3D space. In data analysis, they are used to transform and analyze data in a more meaningful way. In physics, linear transformations are used to describe the motion and transformation of objects in the physical world.