What is a Dual Vector in Bra and Ket Notation?

AI Thread Summary
Bra and ket notation is a fundamental concept in quantum mechanics, where a ket |ψ> represents a column matrix and a bra <ψ| represents a row matrix. The inner product is denoted as <ψ|ψ>, while the outer product |ψ><ψ| results in a matrix. The term "dual vector" refers to the relationship between bras and kets, emphasizing their roles in linear algebra. While matrix notation is useful, the dual vector concept provides a more abstract understanding of quantum states. For further clarity, supplementary resources like the recommended PDF can enhance comprehension of these concepts.
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This isn't a homework problem, but it's so simple that it belongs here.

Can someone please explain to me bra and ket notation? I've been consulting various books and they are all so abstract. Yesterday, my professor told me that a ket |ψ> represents a column matrix and a bra <ψ| represents a row matrix, and that's all there is to it. The (dot) product is represented by <ψ|ψ> rather than <ψ||ψ>, and |ψ><ψ| is also a product, though it produces a matrix.

Yes, this all makes sense in terms of matrices. So my first request is that someone explain this to me in modern, matrix notation.

What's this mysterious "dual vector" that all the books allude to in the introduction of bra and kets? Why isn't matrix notation, ψ for column matrix and ψT for row matrix good enough?
 
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I would recommend you reading this:
www.users.csbsju.edu/~frioux/dirac/dirac.pdf
It helped me a lot and we got it from our QM-teacher as a supplement to the book by Brandsen and Joachain named Quantum Mechanics(Descriptive title)
 
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