Vector Problem. Free and dummy indices?

[P-Jay1]

(ABC)ij =Xk Xl AikBklClj where x is the sum of

In the above equation which are the dummy indices and which are the free indices?

I think i and j are free, and k and l are dummy. But I am not surre!

 

[vela]

That's right.

 

[P-Jay1]

The above is a system of equations. I don't know how many it holds though. Is it 3? How do i show this?

 

[vela]

You have one equation for each possible combination of the free indices, e.g. i=1, j=1; i=2, j=1; etc. How many does that give you?

 

[P-Jay1]

Thankyou vela I see now! I've got one last question:What are the ranges of summation over k and l?

 

[HallsofIvy]

It is impossible to answer that, just as it is impossible to say exactly how many equations that represents, without knowing the dimension of the problem. If two dimensional, all indices range from 1 to 2, if three dimensional, from 1 to 3. If four dimensional, from 1 to 4, etc.