A different convention would make matrix-matrix multiplication no longer express composition of linear maps. So the geometric -- or deeper meaning -- would be l

A different convention would make matrix-matrix multiplication no longer express composition of linear maps. So the geometric -- or deeper meaning -- would be lost. The operation you're describing is nevertheless equal to A^T B. In other words, it can be expressed using a combination of matrix multiplication and matrix transpose. I don't see what it could be used for, though.