Frat boys and matrix elements

First of all, let me just say that I hate frat boys. I really do. Screw frats and the people in them. It's 11 AM and I'm at work at the SLC and there's about 20-30 frat boys here (and they ARE frat boys, you can tell just by looking at them) screaming and running around and cheering. AT THE SLC. For about 2 hours solid. It's intensely obnoxious. Anyone who would be in this area studying has to go to a different area of the building because of these imbeciles. I can't even tell them to shut up because this isn't technically a quiet area.

Anyway...

One thing that turns up over and over in QM2 is 'matrix elements'; this is because you represent operators with linear transformations, and linear transformations are written as matrices. I realized that I don't understand that well why this is, so I'm reviewing that, and using this as my notepad...

For any vector with components a1, a2,..., an where you know what a linear transformation T does to its set of basis vectors, T|a> returns a new vector |b> with components:

b1 = T11*a1 + T12*a2 + T13*a3 + ... + T1n*an
b2 = T21*a1 + T22*a2 + T23*a3 + ... + T2n*an
...
b3 = T31*a1 + T32*a2 + T33*a3 + ... + Tnn*an

And T acting on the basis vectors gives you:

T|e1> = T11|e1> + T21|e2> + ... + Tn1|en>
T|e2> = T12|e1> + T22|e2> + ... + Tn2|en>
...
T|en> = T1n|e1> + T2n|e2> + ... + Tnn|en>

These equations give you the matrix elements for T. Take the inner product of each equation with one of the basis vectors to find these; for example, to find the element T21:

<e2|T|e1> = <e2|T11|e1> + <e2|T21|e2> + ... + <e2|Tn1|en>

But the Tij's are scalars, so you can take them out of the inner products. Assuming this is an orthonormal basis, you are left with only one term on the RHS that is not zero:

<e2|T|e1> = T21<e2|e2> = T21

So this is the formula for finding the matrix elements.