Physics, Quantum Physics, Quantum Mechanics, Quantum, Mathematics.
what object just spits out the same vector when multiplied with a vector? One possibility is the number 1 which should be obvious since 1⋅|V⟩ = |V⟩. However, it actually cannot be 1 since ∑|i⟩⟨i| does not give a scalar value as I will show soon. The second possibility is the identity matrix since