linear operator

A linear operator is a linear map whose domain and codomain are the same space: T : V -> V. That single coincidence — input and output living in one place — is what unlocks the deepest theory in linear algebra. Because you can feed the output of T back in, you can form T(T(v)), and T composed with itself again, and you can ask how T moves V around over and over.

Formally an operator is just a linear map V -> V, so it still obeys T(a u + b v) = a T(u) + b T(v). What makes it special is the set of all of them, written End(V) (for endomorphisms — maps of a thing into itself). Operators are the objects whose eigenvalues, invariant subspaces, and canonical forms (diagonal, Jordan) we spend Volume II classifying. A general map V -> W has no eigenvalues, because v and T(v) live in different spaces and cannot be compared.

Concretely, after fixing one basis of V, an operator becomes a square matrix. Squareness is exactly the algebraic shadow of domain equals codomain. Change the basis and the matrix changes by similarity, but the operator — the geometric action — is the same. The whole game of canonical forms is finding the basis in which that one fixed action looks simplest.

Operators are the central object of operator theory and of much of physics and analysis. Differentiation on polynomials, rotation of the plane, a Markov transition — all are operators, and all yield to the same eigenvalue and invariant-subspace machinery.