A linear operator is a mathematical function that maps elements from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you apply the operator to a sum of vectors or to a scaled vector, the result will be the same as if you applied the operator to each vector individually and then combined the results. Linear operators are commonly used in various fields, including mathematics, physics, and engineering. They can be represented using matrices, making them essential in solving systems of linear equations and in functional analysis. Examples of linear operators include differential operators and integral operators.