1. ## Affine space.

Let X be an affine space in F^n and let Y be an affine space in F^k. We look now on the next Cartesian product:

X x Y = {(x,y)| x in X, y in Y} c F^n x F^k =~F^(n+k)

Prove that if to think on X x Y as subset of F^(n+k) under the identification:

((x_1,...,x_n),(y_1,...,y_k)) =(x_1,...,x_n,y_1,...,y_k)

so we get an affine space.

2. Originally Posted by Also sprach Zarathustra
Let X be an affine space in F^n and let Y be an affine space in F^k. We look now on the next Cartesian product:

X x Y = {(x,y)| x in X, y in Y} c F^n x F^k =~F^(n+k)

Prove that if to think on X x Y as subset of F^(n+k) under the identification:

((x_1,...,x_n),(y_1,...,y_k)) =(x_1,...,x_n,y_1,...,y_k)

so we get an affine space.
What is your question here? what are $\displaystyle X,Y$? are they simply subsets? remember that when you talk about affine space you're simply talking about $\displaystyle \mathbb{F}^n$ for some $\displaystyle n$ without the usual vector space structure. Are you by any chance talking about algebraic subsets (zeroes of polynomials)? Please make a clear question!