What is the difference between the basis of a vector space and the basis of the kernel? I am looking for some clarification on what a 'basis' means for various sets of vectors. In particular the difference, between the basis of a set of vectors and the basis of the kernel.
 A: Given a linear transformation $T : V \to W$, its kernel $\text{Ker}(T)$ is a vector space: $\text{Ker}(T)$  is a subspace of the vector space $V$; and every subspace of a vector space is a vector space in its own right.
So, the definition of basis for a vector space can be applied as it is written to the vector space $\text{Ker}(T)$.
A: There is no difference. Vector spaces can have bases (except in some cases they can't, which is a different discussion entirely), and that's that. Note that this also goes for subspaces of larger vector spaces.
A kernel (of a linear transformation) is a vector space. It's a subspace of the domain (of that linear transformation). And therefore it can have a basis just as much as any other vector space.
Sets of vectors which are not vector spaces do not have bases.
A: The Basis $B$ of a Vector Space $V\ $is any set of vectors in that Vector Space that satisfies these two conditions:
$1)\ $ $V=\text{Span} (B)$
This basically means that every vector in $V$ can be expressed as a linear combination of the vectors in $B$.
$2)\ $ $B$ is $\text{Linearly Independent}$
Loosely put, this states that every vector in $B$ is making its own unique contribution and can not be expressed as a linear combination of the remaining vectors in $B$.


The way I like to think about it is that every Vector Space consists
  of several vectors, but its basis is the smallest possible subset
  of those several vectors which can be used to express every single
  vector in that Vector Space.

The Kernel is a specific kind of Vector Space and everything that I have mentioned above about general Vector Spaces also applies to the Kernel in particular.
