So as I understand i = (1,0,0), j = (0,1,0), k = (0,0,1) span xy plane in space having 0 component for its z
The vectors (1,0,0)^T , (0,1,0)^T and (0,0,1)^T span first of all the [MATH]\mathbb{R}^3[/MATH]. If we say that their z-component is 0, then the system reduces to (1,0,0)^T , (0,1,0)^T respectively (0,0,0)^T.
but we can not say i,j,k span R^2 since R^2 is not a subspace of R^3
Yes!
Saying that span(i,j,k) = xy plane in space is wrong as I understand from your answers.
Yes, thats because we have vecors with three components (x,y,z) and even more the [MATH]\mathbb{R}^2[/MATH] is not a subspace of [MATH]\mathbb{R}^3[/MATH], since it is not even a subset of [MATH]\mathbb{R}^3[/MATH]
Therefore, I am confused now. Could you also explain your ideas about this?
I think you are confused by the word subset here? So we have from the definition:
[MATH]V[/MATH] is a vector space over a field [MATH]K[/MATH] and [MATH]S \subset V[/MATH] is a subspace of this vectorspace, therefore we have [MATH]\text{span}(S) = \left(\sum_{i=1}^{k} \lambda_i v_i | k \in \mathbb{N}, v_i \in S, \lambda_i \in K\right)[/MATH], the span is the set of all finite linear combinations of the [MATH]v_i[/MATH].
A better idea might be to think of the XY plane, the span then contains all possible vectors that can lie in that plane (all possibilities is more or less the span). For example, just imagine your desk, now you can place a pen on it in almost infinite ways. And that's roughly how you can imagine the span. Now you can ask yourself why this is important and what this definition is supposed to do, but at the latest with the concept of the base it is then completely clear why the span is so important. At least this is how I like to think of the span.