In dual basis, why the functions are of the form $\sum_{i=1}^{n}a_ix_i$? My book says (Linear Algebra - Lipschutz): Let V vector space where $dim(V) = n$. Any functional $\phi$ of $V*$ has the representation $\phi(x_1, x_2, ..., x_n) = a_1x_1 + a_2x_2 + ... + a_nx_n$. Why?
I also need to study the topic "Dual basis" in Linear Algebra. What books would you recommend? I am reading Kenneth Hoffman. Thanks in advice. 

Let $X$ be a vector space and $X^*$ be the dual space. Then $X^*$ consists of linear functions $T:X\rightarrow\mathbb{R}$. Let $\{e_i\}_{i=1}^n$ be a basis of $X$. Define $a_i:=T(e_i)$. Any vector  $x\in X$ can be written as $x=\sum_{i=1}^n x_ie_i$. Thus by linearity $T(v)=\sum_{i=1}^n a_i x_i$. 
