So I was reading this wiki entry: https://en.m.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
And stopped specifically at this point
:
So I have a few questions about the last equation. Consider the basis $(X_1,X_2)$ and $(Y_1,Y_2)$ the transformation between them goes as:
$$ Y_1 = a^1_1 X_1 + a^2_1 X_2$$ $$ Y_2 = a^1_2 X_1 + a^2_2 X_2$$
So far, so good, now it is mentioned that $a^i_j$ is a matrix, my doubt is, how does the operation go? $X_i$ $Y_j$ are row or column matrices? The lower índice suggests row, but then the product doesn't make sense unless it is from the right, like $Y_j = X_ia^i_j $ .
What about the indices i,j on the matrix? Which is row and which is column? If the vectors are row, i is row, j is column, if the vectors are column, then i is column, j is row( and the product goes as the formula shows, from the left).
Other wiki entries just make me more confused, where the products go left or right without clarification.