Covariant and contravariat components

[mertcan]

hi, Initially I would like to ask a little and basic question: I know that $$v=\sum_{i=0} e_i v^i$$ where $$v^i=e^i v$$ But sometimes I think we can write the first equation like $$v=\sum_{i=0} e_i e^i v$$, and I am aware that $$e_i e^i=1$$ , then our equation becomes $$v=\sum_{i=0} v$$, ın short if we take the sum over i at the end, our equation becomes v=3v or v=5v...Could you help me get over this confusion I have sometimes in my mind? Very thanks in advance...

 

[Orodruin]

You really should start writing out your inner products or you will keep making mistakes like this. What appears in your third sum is not $(\vec e_i \cdot \vec e^i)\vec v$, it is $\vec e_i (\vec e^i \cdot \vec v)$.

 

[mertcan]

ok I got it I managed to ensure the derivation myself. Thanks...