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If $\{|u_1\rangle, ..., |u_n\rangle \}$ are an orthonormal basis for $\mathbb{C}_n$, then

$$ \sum_{j=1}^{n} |u_j\rangle\langle u_j| = I_n$$

I can see that this is true in the standard computational basis, but I'm having trouble seeing it intuitively when generalized to any basis, nor can I prove it. Can anyone help?

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Let $A$ be the operator give by your sum.

  • Can you see that $A | u_i \rangle = | u_i \rangle$ for each $i$?
  • Can you see why the above observation implies that $A$ is the identity?
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