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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.

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  • $\begingroup$ Think of it as a lego game, where you can build vectors from more "elementary" vectors. The "building rules" are simply + and scaling (i.e., multiplying a vector by a number). A vector space, is a set where taking any subset and "building" using those rules keeps you in it. The "largest" is fuzzy, think of the real line, $\mathbb R$: as you know it is a one-dimensional vector space, if you're a topologist you like to add $\infty$ to it (call the result $\bar{\mathbb R}$): $\bar{\mathbb R}$ is "larger" than $\mathbb R$, a reasonable concept of dimension is $1$, yet it's not a vector space. $\endgroup$ – Oskar Limka Oct 8 '15 at 19:54
  • $\begingroup$ Yes it does define (a better word might be "generate") the entirety of $R^2$, that's what "spans" means. $\endgroup$ – Ned Oct 8 '15 at 20:04
  • $\begingroup$ Are you questioning the difference between a subspace and a space? Any two vectors will (at least) define a subspace, and in this case that "subspace"is the whole space, R^2. So in a pedantic sense, the subspace defined by the vectors does contain the entirety of the space R^2. But that subspace is not different from the space R^2. The vectors both span the subspace and the space (because the subspace is the whole space). (There are no points in R^2 that cannot be "reached" by scalings and sums of your vectors.) $\endgroup$ – dynamo Oct 8 '15 at 20:26
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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 $\{(1,2)(2,1)\}$

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If a set of vectors spans some vector space, it spans the entire space. This is the definition of a spanning some space. With your example it is clear that the two vectors are linearly independent. The geometric representation of their span will then be a plane. This plane can have various orientations, but it will still span all of the two-dimensional Euclidean space.

To clarify, the span of two three-dimensional vectors can be geometrically represented by one of an infinite amount of oriented planes. The two-dimensional case you are mentioning will always be the same oriented plane. Think of the span of two linearly independent (2-D) vectors as your standard coordinate plane.

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