Fields, sets and vector spaces 
How do I show this? I am new to all concepts in this question, however I am aware of the definitions of all concepts given.
 A: You have a definition of vector space. It is a set (in the problem, $V$ is this set), and a pair of operations.


*

*Figure out what operations are these, and how they are defined. It is easy, don't bother with bizarre operations.

*Prove that these operations satisfy all the properties that are required in the vector space definition. (This completes the first question).

*Name the elements of the set $S$, for example $\{x_1,\ldots,x_n\}$. Try to find a base for $V$. $1$'s and $0$'s can be very handy. Prove that this is indeed a base.

*How many elements has this base? This is the dimension of $V$. And you are done.

A: Let $W$ be the set of functions from $S$ to $K$.  
First you need to define addition and scalar multiplication on $W$.  If $f$ and $g$ are functions from $S$ to $K$ we define $(f+g)(x)=f(x)+g(x)$ and for $\lambda$ in $K$ we define $(\lambda f)(x)=\lambda(f(x))$.  Now you must check that the axioms of a vector space are satisfied.  
For the second part, let $S=\{s_1,\dots,s_n\}$ and show that a basis of $W$ is given by the functions $f_i$ for $i=1,\dots,n$ where $f_i(s_j)$ equals $1$ if $i=j$ and $0$ otherwise.  
