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**Original Question (Please ignore this):**

I knew this when I read about it the first time a while back but can't put the two and two together anymore. :)

I understand ##\vec{a_i}\centerdot \vec{a^j}=\delta_i^j## and ##g_{ij}=\vec{a_i}\centerdot\vec{a_j}##

How does one get from there to ##\vec{a_i}=g_{ij}\vec{a^j}##?

**Modified Question (Please answer this):**

How can you prove that the metric tensor can be used to raise and lower indices on any tensor? How does the metric help transform between covariant and contravariant tensors? Proving this for rank one tensors should be good enough. How does the metric perform the conversion ##T_{i}=g_{ij} T{^j}##?

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