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Given any unit vector, $v\equiv |v\rangle\in\mathcal X$ for some finite-dimensional complex vector space $\mathcal X$, the operator $vv^\dagger\equiv|v\rangle\!\langle v|$ defined by $$|v\rangle\!\langle v|\equiv v v^\dagger\in \operatorname{Lin}(\mathcal X), \\ (vv^\dagger)(w)\equiv (|v\rangle\!\langle v|)(|w\rangle) \equiv v \langle v,w\rangle,$$ is a rank-...
Background If $v_1, v_2, \dots, v_n$ is an orthonormal basis in the inner product space $V$, then any vector $u\in V$ can be expressed as a linear combination $$u = \alpha_1 v_1 + \alpha_2 v_2 + \dots + \alpha_n v_n.\tag1$$ Moreover, the coefficients can be computed using $\alpha_k=\langle v_k, u\rangle$, as can be seen by applying $\langle v_k, .\rangle$ ...