- #1

- 244

- 32

Let's say I choose to index matrices like: superscript denotes row number, subscript denotes column number. Let's try to apply this in change of basis equations for vectors and covectors. Suppose for a vector space ##V##, we switch from basis ##H## to ##\tilde H##. The corresponding dual bases are ##\Theta## and ##\tilde\Theta##. If ##b=[id_V]^H_{\tilde H}##, then $$\tilde h_j=b^i_jh_i$$ and indeed ##b^i_j## is the ##i,j##-th element of ##[id_V]^H_{\tilde H}##, since the ##j##-th column of this matrix is the representation of ##id_V(\tilde h_j)## in the ##H## basis. Given some vector ##v##, multiplying both sides by ##\tilde v^j##, we get $$v=b^i_j\tilde v^jh_i\implies v^i=b^i_j\tilde v^j\implies v(\theta^i)=b^i_jv(\tilde\theta^j)=v(b^i_j\tilde\theta^j)\\

\implies \theta^i=b^i_j\tilde\theta^j$$ The issue is that in THIS case, I cannot interpret ##b^i_j## as the ##i,j##-th element of ##[id_V]^H_{\tilde H}##, since the coefficients in the RHS are supposed to represent the column elements of the CoB matrix. In other words ##j## represents the row number this time, and ##i## represents the column number!

Does any consistent notation exist for tagging row/column indices in matrices in this whole treatment? Something that holds true for both the CoB equations for ##\tilde h_j## and ##\theta^i##? I have a vague idea about some staggered notation, but even if I were to stagger the subscript a bit to the right in all equations above, it would still give inconsistent results.