What is the role of the projector operator in vector manipulation?

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The discussion centers on the role of the projector operator in vector manipulation, specifically regarding when to halt computations. The user has reached the expression ## <k|j> \sum_{j} c_{j}|k>##, indicating that the projector operator effectively projects the state onto |k>. They note that the coefficients have changed, leading to confusion about the next steps in the process. The simplification of the summation to a single component occurs because <k|j> equals 1 only when j equals k, and zero otherwise. Clarification on the implications of these changes and the subsequent actions required is sought.
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I don't know what is the answer, so i am not sure when to stop the computation or not.
The far i reached was ## <k|j> \sum_{j} c_{j}|k>##. That is, the action of the projector operator is, obviously, project the state in |k>. Now, the coefficients was changed. So now what i have to do?
 
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You see <k|j>=1 for j=k otherwise zero so the summation is reduced to a single component.
 

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