It's related to the Three Rules of Linear Algebra:
1) Don't pick a basis.
2) DON'T pick a basis.
3) If you do pick a basis, choose very carefully.
The point is, vectors are not lists of numbers. The list of numbers, together with a choice of basis, does determine a vector. But change the basis and your numbers change. Much of the stuff you need to do with linear algebra does not depend one bit on the basis.
For instance, you can compute the rank of a matrix by doing row-echelon reduction. Change the basis, and you get what looks like an entirely different matrix, but if you compute its rank, it's the same. That's because the two matrices represent the same linear transformation, under two different bases. And the rank is really a property of the transformation, so it makes sense that it's the same for every representation of that transformation.
Likewise, vectors have properties that are independent of the chosen basis. This question is trying to get you to think about vectors as objects in their own right, and to consider what properties they have that are independent of your choice of basis.