Linear Algebra, 5th Edition

6.3 The Adjoint of a Linear Operator

In Section 6.1, we defined the conjugate transpose A* of a matrix A. For a linear operator T on an inner product space v, we now define a related linear operator on V called the adjoint of T, whose matrix representation with respect to any orthonormal basis $\beta$ for V is ${[\text{T}]}_{\beta }^{\text{*}}$. The analogy between conjugation of complex numbers and adjoints of linear operators will become apparent. We first need a preliminary result.

Let V be an inner product space, and let $y\in \text{V}$. The function $g:\text{V}\to F$ defined by $g(x)=\langle x,y\rangle$ is clearly linear. More interesting is the fact that if V is finite-dimensional, every linear transformation from V into F is of this form.

Theorem 6.8.

Let V be a finite-dimensional inner product space ...