A linear map is a mathematical function that transforms vectors in a way that preserves the operations of vector addition and scalar multiplication. This means that if you have two vectors, u and v, and a scalar c, a linear map T satisfies the properties: T(u + v) = T(u) + T(v) and T(cu) = cT(u). Linear maps are often represented by matrices in linear algebra. Linear maps are essential in various fields, including physics, computer science, and engineering, as they help model systems and transformations. They can be used to simplify complex problems by breaking them down into linear components, making them easier to analyze and solve.