Dual space of vector space spanned by the vector $[1, 0]'$ in $\mathbb{R} ^2$

Question

From what I understand, the dual space of a finite vector space is unique and has the same dimension as the original vector space. However, my confusion arises when we consider the vector space spanned by the vector $[1, 0]'$ in $\mathbb{R} ^2$. According to the definition of the dual space, any function of the form $f(u) = [a, b]' \cdot u$ where $a$ and $b$ are arbitrary Reals (assuming the field where we are mapping the vector space is $\mathbb{R}$) will give a linear map. Thus, the dual space should be a two-dimensional vector space, since $a$ and $b$ are essentially independent. Can anyone please point out where does my argument fall apart?