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

[ThereIam]

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

 

[DrClaude]

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