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

[LCSphysicist]

Homework Statement

\\

Relevant Equations

\\

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?

 

[anuttarasammyak]

You see <k|j>=1 for j=k otherwise zero so the summation is reduced to a single component.