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A linear operator $\hat{A}$ on a vector space is a function that maps vectors to vectors, $|\psi \rangle \rightarrow \hat{A}|\psi\rangle$ such that \[\hat{A} \left ( a |\psi \rangle + b|\phi \rangle \right ) = a\hat{A} |\psi\ket + b\hat{A} |\phi \ket .\]

Given a linear operator $\hat{A}$ on an inner product space, we can also define an action of a linear operator on a dual vector. The dual vector $\langle \psi | \hat{A}$ is defined to be the unique dual vector such that \[\left ( \langle \psi | \hat{A}\right ) |\phi \rangle = \langle \psi | \left ( \hat{A} |\phi \rangle \right ), \] for all vectors $|\phi \rangle$.

Note that, given this definition, an expression like $\langle \psi | \hat{A} | \phi \rangle$ is unambiguous. We will always get the same result regardless of whether we first apply $\hat{A}$ to $|\phi \rangle$ and then take the inner product with $|\psi \rangle$, or first apply the operator $\hat{A}$ to the dual vector $\langle \psi |$ and then apply it to $|\phi \rangle$.