The connection between span and space? I have $L=\{p(x)=ax^3+bx^2+cx+d \in R_4 | a=2b \mbox{ and } c=d\}$ over $\mathbb R$.
Now I need to find the span of it and prove by it that this $L$ is a space over $\mathbb R$.
So by playing with it I have found that the span is $\{2x^3 + x^2, x+1\}$.
Judging by what we did in class now I would just say "..and therefore $L$ spans $R_4$"
But I don't understand why? I can always find a span for any group of polynomials so does it mean that polynomials will always span $R_4$? I am confused about the connection between spans and spaces.
Thanks in advance 
 A: So, let's start with the definitions. The span of a set $S=\{v_1,v_2,\dots,v_r\}$ is the set of all linear combinations of the elements of $S$, that is, it is the set of all vectors of the form $$c_1v_1+c_2v_2+\cdots+c_rv_r$$ where $c_1,c_2,\dots,c_r$ are real numbers. It is easy to check that the span of any set is nonempty and is closed under addition and under multiplication by scalars, so the span of any set is a vector space. 
Conversely, given any vector space $V$, there is always a set $S$ such that the span of $S$ is $V$. Indeed, you could just take $S$ to be $V$. But if $V$ is a subspace of ${\bf R}^n$ for some $n$, then there will be sets $S$ with at most $n$ elements such that the span of $S$ is $V$. 
Now, in your example, you are talking about vector spaces whose elements are polynomials. So presumably your $R_4$ is the vector space of polynomials of degree at most three. This is not the same as ${\bf R}^4$ (since the elements of ${\bf R}^4$ are 4-tuples, not polynomials), but it is isomorphic to ${\bf R}^4$. 
Now it is correct that if $S=\{2x^3+x^2,x+1\}$ then the span of $S$ is your vector space $L$. Indeed, as soon as you know that the span of $S$ is $L$, you know that $L$ is a vector space (by the first paragraph of this answer). But of course $L$ is not all of $R_4$. For example, $x$ is in $R_4$, but not in $L$. 
I hope this clarifies things, and I hope you'll think it through until you understand it. If you have further questions after you have thought it through for a while, ask. 
