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.

You have a definition of vector space. It is a set (in the problem, $V$ is this set), and a pair of operations.


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*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.

