(Sorry if this is in the wrong forum, wasn't sure where to post)

So I'm taking continuum mechanics and trying to get my head around the notation etc. We're using the einstein summation notation where a repeated index implies summation over that index between 1 and 3. In my class notes on change of basis we have the following (anything in red is an index):

To change basis vectors, the new basis vectorse'i can be written as a linear combination of the old basis vectorsej (i, j = {1, 2, 3}) such that:

e'i = lijej......(1)

Where lij are the direction cosines. Taking dot product withej on both sides gives:

lij =e'i.ej .....(2)

Also, the kronecker delta d can be written:

dij =e'i.e'j = likeke'j = lik ljk (using (1) and (2))

That is:

dij = lik ljk .....(3)

So that's what the class notes say. But what confused me is when I tried to write (3) out in full, summing the RHS over k, I got this:

dij = li1 lj1 + li2 lj2 + li3 lj3

Now using (2) :

dij = (e'i.e1)(e'j.e1) + (e'i.e2)(e'j.e2) + (e'i.e3)(e'j.e3)

dij =e'i.e'j +e'i.e'j +e'i.e'j

dij = 3dij

Um, yeah. 3 = 1 ...

Obviously I've gone totally wrong somewhere. Can anyone point me in the right direction? o_O;