In this unit, we will delve into the concepts of linear operators and functionals, which are fundamental to the study of vector spaces and their applications in various fields.
A linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, if T is a linear operator, T(x + y) = T(x) + T(y) and T(cx) = cT(x) for all vectors x and y and all scalars c.
The null space of a linear operator T, denoted by N(T), is the set of all vectors x such that T(x) = 0. The range of T, denoted by R(T), is the set of all vectors that can be obtained by applying T to some vector in the domain.
A linear operator T is said to be injective (or one-to-one) if different vectors in the domain always map to different vectors in the range. T is surjective (or onto) if its range is the entire codomain.
If T is a bijective linear operator, then there exists an inverse operator T^-1 such that T^-1(T(x)) = x for all x in the domain of T, and T(T^-1(y)) = y for all y in the range of T.
A linear functional is a linear operator whose range is the field of scalars. The set of all linear functionals on a vector space V is itself a vector space, known as the dual space of V, and is denoted by V*.
Given a linear operator T between two inner product spaces, the adjoint of T, denoted by T*, is a linear operator such that <Tx, y> = <x, T*y> for all x and y in the domain of T.
In the next unit, we will explore how these concepts extend to topological vector spaces, where the topology allows us to consider concepts of continuity and limit.