Changing bases (with bra-ket notation)

[iScience]

http://i.imgur.com/ORtBJdT.jpg

i don't understand why the old base is written in terms/as a linear combination of the new bases. wouldn't i want to map my coordinates from old to new not new to old?..

here's what my textbook says about it, can you guys interpret this for me, i still don't get it..

(it's the stuff of the red outline)

http://i.imgur.com/n5HJ7F2.jpg

 

[jcsd]

Let's say you have an arbitrary vector written in terms of the old basis. Is it easier to write it in terms of the new basis knowing what each of old basis vectors are in the new basis, or is it easier to start by knowing what the new basis vectors are in the old basis?

 

[iScience]

i don't follow; how are the two any different?

in both cases it sounds like I'm writing the new basis in terms of the old. can you elaborate on what you mean?

 

[jcsd]

Given a vector:

|v\rangle = \sum^n_{j=1} x_j|e_j\rangle

and that

|e_j\rangle = \sum^n_{i=1} S_{ij}|f_i\rangle

then

|v\rangle = \sum^n_{j=1}\sum^n_{i=1}x_j S_{ij}|f_i\rangle

Or in other words knowing how to express the old basis vectors in terms of the new basis vectors tells you how to express a vector in the new basis given it's expression in the old basis.