Regarding standard basis and coordinate vectors Suppose we have the standard basis of R^2 i.e (1,0) and (0,1). Then the coordiate vectors and basis  are in the same set. Is this right? The coordinate vectors should be distanct from the vector space. I.e in the case of R^2 we should have two distinct but identical sets
 A: A basis and coordinates with respect to a basis are related. I'm going to try to answer the question as I interpret it - but I still feel like this question may be closed as unclear.
So a basis $\mathcal{B} = \{ v_1, \ldots, v_n \}$ is a linearly independent set in $V$ that spans $V$. We can represent any vector $v \in V$ as a linear combination of the basis elements:
$$ v = \sum_{j = 1}^n c_j v_j $$
and we can call the coordinates of $v$ the weights $c_j$. That is, we can write
$$ v = (c_1, \ldots, c_n)_\mathcal{B}.$$
It just so happens that this coordinate is itself an n-tuple. However, this should be interpreted as a collection of weights, each of which tells you how many "spaces" to move in the direction of $v_j$. Compare this to the standard basis of $\mathbb{R}^2$, and consider the vector $(3, 4)$. We move 3 "spaces" in the direction of $(1, 0)$ and 4 in $(0, 1)$. Again, it so happens that $(3, 4)$ is a vector in $\mathbb{R}^2$ but this gives the weights of the associated linear combination when we expand in terms of the basis.
I hope this helps.
