# Definition Consider the [[vector space]] of [[Linear Operator|linear operators]] on a vector space $W$, denoted $\mathcal{L}(W)$ (see [[Linear Operator#Vector Space|here]]). What if we set $W = \mathcal{L}(V)$ for some other [[vector space]] $V$? This is definitely possible since $\mathcal{L}(V)$ is also a vector space. In this case, $\mathcal{L}(\mathcal{L}(V))$ is the vector space of linear operators on linear operators, i.e., the elements ([[vector|vecotrs]]) in this vector space take linear operators into linear operators! Given two operators on $V$, i.e., $A, B \in \mathcal{L}(V)$, then we can define the *adjoint action* or *adjoint representation* as follows: $ \text{ad}:\mathcal{L}(V) \times \mathcal{L}(V ) \rightarrow \mathcal{L}(V) $ such that: $ \text{ad}_A(B) \equiv [A, B] $ where $[.,.]$ indicates the [[commutator]]. # Applications in Physics The [[Heisenberg Picture|Heisenberg picture]] in quantum mechanics emphasizes $\mathcal{L}(V)$ instead of $V$, and interprets the [[Hamiltonian]] as an operator in the adjoint representation. The [[Heisenberg Picture|Heisenberg equation of motion]] for any observable $A$ reads: $ \frac{dA}{dt} = i \text{ad}_{H}(A) $