A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you take two vectors and add them together, the transformation of that sum will be the same as transforming each vector individually and then adding the results. Linear transformations can be represented using matrices, which are rectangular arrays of numbers. When a vector is multiplied by a matrix, the result is a new vector that reflects the transformation. Common examples of linear transformations include rotation, scaling, and shearing.