Because it tells you how to go from local time to UTC — and a change-of-basis transformation A from old to new is defined to be such a transformation that x_old

Because it tells you how to go from local time to UTC — and a change-of-basis transformation A from old to new is defined to be such a transformation that x_old = A x_new , not the other way around, x_new = A x_old . So, you have 02:00 in some local time, and you want to go to UTC? The change-of-basis transformation is "add 1 hour", because 01:00 (in UTC) plus 1 hour results in 02:00 (in local time), all perfectly reasonable, from an algebraist's point of view.