With a linear transformation the relationship between pairs might change. For example A1 and A2 might be far apart, but this implies that B1 and B2 are also far apart. The distance between the pairs might shrink, but if it does, then both B1 and B2 will be close to A1 and A2 at the same time. We can apply many linear transformations, but the relationship between A1 / A2 and B1 / B2 will always be similar.