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If we express the projection operator with vectors, we get ##\hat{P}\vec{v} = \vec{e}(\vec{e}\vec{v})## which means that we project ##\vec{v}## onto ##\vec{e}##. We can write this as ##\hat{P}\vec{v} = e_k \sum_{l} e_lv_l = \sum_l (e_ke_l )v_l##. In my class we said that the matrix for the projection operator is ##P_ {kl}=e_ke_l##, so ##\hat{P}\vec{v}=\sum_l P_{kl} v_l##. But isn't ##e_ke_l## equal to ##\delta_{kl}##?