What is a Dual Vector in Bra and Ket Notation?

[Harrisonized]

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 ψ^T for row matrix good enough?

 

[dikmikkel]

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)