Not a Vector

A list of numbers is not a vector. If you copy down the three winning lottery numbers and plot them in inches and then you plot them in centimeters, the vector will shrink.

Being just a list of numbers, they lack the intelligence to do the right thing and change in a way to undo the effect of the change to the coordinate system.

They scale with the coordinate system and they rotate with coordinate system.

10 inches becomes 10 centimeters.

But if out in the math wilderness you encounter a critter that goes from 10 in to 25.4 cm, you recognize that it is intelligent.

Appendix A

Another issue for us to recognize. It may be that because you are being introduced to this stuff, we work with simple systems and in those systems any list of numbers can qualify to be a vector. It may be that we don’t run into issues until we go to more complex spaces and coordinate systems where it is possible to write down numbers that won’t do what needs to be done.

We may have found the treasure trove: the secret may be that if a transformation matrix is orthogonal then vectors and one forms will transform the same way.