A set with co-dimension less continuum Define the dimension of a vector space to be the cardinality of any basis for the vector space. Also every subspace $S$ of $\mathbb R$ also has a codimension, which is the dimension of $\mathbb{R}/S$, or equivalently the dimension of any complementary subspace to $S$. My question is not hard, let $S$ be a linear independent subset of $\mathbb R.$ I want to consider two cases:
(1) If $S$ has co-dimension$<|\mathbb R|.$
(2) If $S$ has c0-dimension equal to $|\mathbb R|.$
Is it true for both cases that $S$ would not be a basis, that is, we can not use   Zorn's lemma to extend it to a basis. Or It is just true for (2).
Any help would be appreciated greatly
 A: EDIT: Per the comments below, it seems I misunderstood the question. I've edited my answer accordingly.

Whether or not the codimension is of size continuum isn't the issue: rather, the answer is determined entirely by whether the codimension $codim(span(S))$ is nonzero.
The codimension of a subspace $B$ of a vector space $V$ is a measure of the "smallness" of $B$ from the point of view of $V$: the higher the codimension, the more stuff in $V$ that's not in $B$. Now codimension is defined in terms of the quotient space $V/B$, but it is still related to the difference set $V\setminus B$ (gotta love that notation). Specifically, $codim(B)>0$ iff $B\not=V$:

*

*Since $V/V$ is the one-element vector space, $B=V$ clearly implies $codim(B)=0$.


*In the other direction, let $a$ be a nonidentity element of $V/B$. This $a$ is an equivalence class of elements of $V$, so let $\hat{a}\in V$ be a representative of $a$. Since $a$ is not the identity in $V/B$ the representative $\hat{a}$ must not be equivalent modulo $B$ to the identity vector of $V$ - that is, we must have $\hat{a}-{\bf 0}_V\not\in B$. But this is just a silly way of saying $\hat{a}\not\in B$. In particular we have $V\setminus B\not=\emptyset$ and so $B\not=V$.
Now beyond that there isn't too tight a relationship between the codimension $codim(B)$ and the size of the difference set $\vert V\setminus B\vert$. For example, let $B_1$ and $B_2$ be $1$- and $2$-dimensional subspaces of $\mathbb{R}^3$ as an $\mathbb{R}$-vector space, respectively. Then their codimensions are different (specifically $codim(B_1)=2$ and $codim(B_2)=1$) but the difference sets $\mathbb{R}^3\setminus B_1$ and $\mathbb{R}^3\setminus B_2$ each have cardinality continuum. So the OP is a very special circumstance: we get a connection between the codimension and the difference set only because the question we're asking is particularly "coarse."
