I'm looking for conceptual answers. From my understanding a vector space is the "largest" space possible for it's dimension. Example: $R^2$ contains all possible 2-vectors. And, from my understanding, given a set of 2-vectors they will only span $R^2$ if they are linear independent. Example: $\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\}$ spans $R^2$
But does it span all of $R^2$? It, by definition, is a subspace of $R^2$, but can't this set also define the entirety of $R^2$? Why or why not?
I realize I'm probably not asking the right questions and have some sort of fundamental misunderstanding so please try to point out my conceptual errors, rather than the classic professorial reply, "I can't answer that question because it doesn't make sense." Thank you.
Best Answer
Ok, so you have the set $\{(1,2)(2,1)\}$ which has exactly two elements. So it definetely is not $\Bbb{R}^2$. However it does span $\Bbb{R}^2$ what does this mean? Well, that given ANY vector $(v_1,v_2)\in \Bbb{R}^2$ there exist $a,b \in \Bbb{R}$ such that $a(1,2) + b(2,1)=(v_1,v_2)$. Check it out yourself!
So the essential difference is that when you add the word span to the sentence, everything changes, in fact,
$$span((1,2)(2,1))=\{a(1,2)+b(2,1) : a,b \in \Bbb{R}\}=\Bbb{R}^2$$
Take out the word span, and you have just a simple to element subset of $\Bbb{R}^2$, say $\{(1,2)(2,1)\}$