I am trying to develop a feeling for duality and stumbled upon the following question:
Say we are in $\mathbb R^2$ and let $U=\operatorname{span}([2,1]')$ be our subspace. My question is: what is $U^*$ (the dual space of $U$)?
I assume that since $\dim V = \dim V^*$ for any finite-dimensional vector space (and $U$ is one) $U^*$ should be a line as well, a line in $(\mathbb R^2)^*$ with row vectors so to speak. Yet it looks like anything from $(\mathbb R^2)^*$ would go: $ \varphi: U \to \mathbb R, [x,y]' \mapsto [3,4]*[x,y]'$ seems to be linear as well and so by definition $ \varphi \in U^* $. So whatever we have instead of $[3,4]$ the function is still linear.
And how should we construct the dual basis for $[2,1]'$? I understand the $\delta_{ij}$ way when we are dealing the the dual space of a "true" vector space (meaning it's not a subspace of any other vector space except itself).