I am examining a space (in $52$ dimensions). I found $3$ bases in it: $A$, $B$ and $C$. The conversion matrix of $A$ to $C$ is upper triangular with only $0$ or $1$ entries and determinant 1. The conversion matrix of $B$ to $C$ is the transpose of the first. What does this tell me about $A$ and $B$ (if anything)?
tx.
I'm afraid not much. If I understand you correctly, you have $C = A \cdot U$ and $C = B \cdot U^T$. Since $U$ is invertible, this means $A = B \cdot U^T \cdot U^{-1}$. However, there is nothing really special about $U^T U^{-1}$: For an upper triangular $U$, its transpose is lower triangular and its inverse is upper triangular (both have all ones on their main diagonal). However, the product of an upper and a lower-triangular matrix is full and pretty arbitrary. Also, while $U$ is binary, $U^{-1}$ isn't. It is an integer matrix (since the determinant is equal to one) but its values can be positive and negative and much larger than one.
The only thing you do know is that the first column of $A$, $B$, and $C$ agree entirely, since $U$ must have a single one in its top-left corner.