Spanning Theory of Linear Algebra I am not really sure how we can decide that some linearly independent set spans $\mathbb R^n$? Can you explain what are the criteria when we say it spans $\mathbb R^n$?
 A: If it spans a space, then the elements of that space can be written as the linear combination of its elements.
Lets assume that $<1,2>,<3,4>$ are two vector. If we can the values of $c_1,c_2$ in terms of $a,b$ when we wrote it in the form of $c_1<1,2>+c_2<3,4>=<a,b>$ ,then it spans the space.
In other words, when you write a linear equation in the form of augmented matrix, if you have infinity many solutions then it spans the space.
When we comes to your question , if we cannot find $b$ in the system of $Ax=b$, then it does not span the space
A: A "spanning set" or "a set that spans the vector space" is a set such that any member of the vector space can be written as a linear combination of vectors in the set.
A spanning set is NOT necessarily a basis.  A basis is a set that both spans the vector space AND such that the vectors in the set are linearly independent.  It easy to see that a set consisting of a single (non-zero) vector is linearly independent but probably wont span the space!  As we add more vectors to the set they are less likely to be independent but more likely to span the space.  On the other hand, the set of all vectors certainly spans the space but can't be linearly independent.  As we remove vectors from the set they become more likely to be independent but less likely to span the space.
In Linear Algebra, dealing with finite dimensional vector spaces, there is one number, n, where they "meet in the middle".  That is, there exist a set of vectors that both spans the space and is linearly independent.  That set is a basis for the space and the specific number "n" is the "dimension" of the vector space.
