Can two bases for the same vector space have a different number of vectors?
No. The "basis" with more vectors must be linearly dependent and thus not a basis.
What is the purpose of an interface?
An interface describes a general contract to which a class must adhere when implementing the interface. For example, there could be many different types of car that all implement the same car interface which could involve methods such as turn() and accelerate().
What is a lower bound time complexity on a comparison based sorting methods?
O(nlogn) which is achieved by algorithms such as mergesort and heapsort.