Understanding Dirac notation - Product of ops. is product of matrices

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The discussion centers on understanding Dirac notation and its application to the product of operators as matrices. The author, Shankar, illustrates that the product of two operators corresponds to the multiplication of their matrix representations. A key equation involves expressing the product of operators in terms of their matrix elements, specifically relating the inner products to sums over intermediate states. The participant seeks clarification on a specific part of the proof involving the summation over states. Overall, the thread emphasizes the connection between operator products and matrix multiplication in quantum mechanics.
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Homework Statement


This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof.

Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors.

Homework Equations



(Ωλ)_{ij} = <i|Ωλ|j> = <i|ΩIλ|j> =

(and this is the part I don't understand)

\sum(over k) <i|Ω|k><k|λ|j> = \sum (over k) Ω_{ik}λ_{kj}

I apologize if the formatting didn't work. I'll try to fix it asap.

The Attempt at a Solution

 
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Could you be more precise as to what you don't understand?
 

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