How is a linear transformation described in linear algebra?

A linear transformation in linear algebra is fundamentally defined as a mapping that preserves the operations of vector addition and scalar multiplication. This means that if you have a linear transformation ( T ), for any vectors ( u ) and ( v ) and any scalar ( c ), the following conditions must hold true:

1. ( T(u + v) = T(u) + T(v) ) (the transformation of the sum of the vectors is equal to the sum of their transformations).

2. ( T(cu) = cT(u) ) (the transformation of a scalar multiple of a vector is equal to the scalar multiplied by the transformation of the vector).

This preservation of the vector space operations ensures that the structure of the space is maintained under the transformation, which is a critical characteristic of linear transformations. Understanding this definition is essential for analyzing various properties of linear maps, such as their kernel and image, as well as determining whether a transformation is linear or not.