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 a linear 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 represented in the context of linear algebra and can be expressed using matrices. They play a crucial role in various fields, including quantum mechanics, signal processing, and differential equations, where they help simplify complex problems by transforming them into more manageable forms.