Difference between span and basis What is the difference between the span of the image of a matrix and the basis for the span of the image of a matrix? Are these the same thing?
 A: A spanning set for a space is a set of vectors from which you can make every vector in the space by using addition and scalar multiplication (i.e. by taking "linear combinations").
For example in $\mathbb{R}^2$ the three vectors $(1,0),(0,1)$ and $(1,1)$ form a spanning set. I can make the vector $(x,y)$ by doing $x(1,0)+y(0,1)$.
A basis for a space is a spanning set with the extra property that the vectors are linearly independent. This essentially means that you can't make one of the vectors in the spanning set out of the others. In other words a basis is a kind of most efficient spanning set, there are no vectors in our spanning set that weren't needed.
For example in $\mathbb{R}^2$ our spanning set above is not a basis since $(1,1)$ is redundant in the span, we could already make it with $(1,0),(0,1)$. However the set $\{(1,0),(0,1)\}$ is a basis since we cannot discard any more vectors and still span the plane.
A: In general, iff a set of vectors is linearly independent it's called a basis for the span of these vectors.  That's all. 
Edit:  Let's clarify some common phrases.  A basis is always a basis for ... In that sense a set of linear independent vectors is a basis for the span of that set of vectors. 
A: Span is nothing just but all the linear combinations of the vector. Suppose you have a matrix $A$ and you know its image now you woould like to know whether the image can be spanned by finite vectors or not, Suppose It did  and you find the linear independent vectors among them which spans the whole image and that set would be your Basis. 
Basically span of the image would end up giving you the image and the span of the basis of image would give you image of $A$  
A: While a span is always a subspace and not linearly independent, a basis may not be a subspace and is always linearly independent. 
A: Here is a lecture from Gilbert Strang (MIT) about Span and Basis. His explanation is very good, so posting the video here instead of instead of answering it again. All the sessions around Linear Algebra can be found here.
