i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of the form $Z(f_1,\ldots,f_n)$ where $f_i$ are polynomials of degree 1." (from notes by Lothar Goettsche: http://users.ictp.it/~gottsche/)
My problem with this statement is the following: For every point $a =(a_1,\ldots, a_n)$ in $A^n$ i can give a set $S$ of polynomials such that their common zero set ist $\{a\}$: $S = \{x_1-a_1, \ldots, x_n - a_n\}$. But how can i find a set of polynomials that vanishes on a whole vector space ? Of course one can take the zero polynomial, but that would not be of degree one ?
Thank you for advice
readingframe