Einstein Notation: Xj to Xi Transformation Equations

[Gear300]

From what I'm reading, X^j = a_i^jXⁱ (X is a vector and the subscript for a is i)...though I'm not sure where the a_i^jXⁱ came from. Would this by any chance be a relation for the transformation equations from X^j to Xⁱ?

 

[HallsofIvy]

Yes, $x^j$ and $\underline{x}^i$ are vectors. $a_i^j$ is a mixed tensor of order 2. It's not clear to me what you mean by "given a vector summation" but on the right side of the equation there certainly is a summation: The "Einstein convention" is that when an index is repeated, once as a subscript and once as a superscript, you sum over all values of that index.

 

[Gear300]

I see. Thanks for the assistance.