Task interpretation for Hoffman Kunze Linear Algebra exercise 1 (b) sec. 3.6 
I don't understand the task from (b). Is it equivalent to : for every linear functional $f(x_{1}, ..., x_{n})=c_{1}x_{1}+...+c_{n}x_{n}$ on $F^{n}$ which satisfy $c_{1}+...+c_{n}=0$ there exists exactly one and unique functional that belongs to the dual space of $W$?
If it is so, could someone please give me a hint?
 A: Yes. That is, for each $(c_1,\ldots,c_n)\in W$, define 
$$f_{(c_1,\ldots,c_n)}(x_1,\ldots,x_n) = c_1x_1 + \cdots + c_nx_n$$
and let $W'= \{f_{(c_1,\ldots,c_n)} : (c_1,\ldots,c_n)\in W\}$. Then $W'$ is a subspace of $(F^n)^\ast$.
Part (b) asks to prove that there exist an isomorphism between $W^\ast$ and $W'$.
Clearly $W$ and $W'$ are isomorphic. Can you take it from here?
A: I worked out the details from leo's hint. 
Consider the sequence of functions
$$W\rightarrow (F^n)^{*} \rightarrow W^{*}$$
where the first function is
$$(c_1,\dots,c_n)\mapsto f_{c_1,\dots,c_n}$$
where $f_{c_1,\dots,c_n}(x_1,\dots,x_n)=c_1x_1+\cdots c_nx_n$ and the second function is restriction from $F^n$ to $W$
We know both $W$ and $W^{*}$ have the same dimension.  Thus if we show the composition of these two functions is one-to-one then it must be an isomorphism.  Suppose $(c_1,\dots,c_n)\in W\mapsto  f_{c_1,\dots,c_n}=0\in W^{*}$.
Then $\sum c_i=0$ and $\sum c_ix_i=0$ for all $(x_1,\dots,x_n)\in W$.
In other words $\sum c_i=0$ and $\sum c_ix_i=0$ for all  $(x_1,\dots,x_n)$ such that $\sum x_i=0$.
Let $\{\alpha_1,\dots,\alpha_{n-1}\}$ be the basis for $W$ from part (a) ($\alpha_i$ has one in the first component and $-1$ in the $i$-th component).  Then $f_{c_1,\dots,c_n}(\alpha_i)=0$ $\forall$ $ i=1,\dots,n-1$; which implies $c_1=c_i$ $\forall$ $i=2,\dots,n$.  Thus $\sum c_i=nc_1$.  But $\sum c_i=0$, thus $c_1=0$.  Thus $f_{c_1,\dots,c_n}$ is the zero function.
Thus the mapping $W\rightarrow W^{*}$ is a natural isomorphism.  We therefore naturally identify each element in $W^{*}$ with a linear functional $f(x_1,\dots,x_n)=c_1x_1+\cdots c_nx_n$ where $\sum c_i=0$.
