"Usefulness" of Basis for a Vector Space, General?
I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension"
a student asked me what was the use or purpose of a basis for a vector space V.
All I could think of is that bases allow us to define a linear map L for all vectors, once
we know the value of L at the basis vectors for V, i.e., vector spaces are free in their
bases and so on. I mumbled something about identifying all vector spaces over the
same field by their dimension, i.e., if V,V' v.spaces over F both, with the same dimension,
then they are isomorphic.
Are there other aspects where bases are equally important or more?