I'm looking at practice problems for a PhD exam, and I'm uncertain on how to approach the last part of this one.
Let $V$ be $\mathbb{R}^n$ equipped with the standard inner product. For an arbitrary subspace $U$ of $V$, let $U^\perp = \{v \in V | \langle u,v \rangle = 0 \text{ for all } u \in U\}$.
(a) Show that $U \cap U^\perp = \{0\}$
(b) Show that $U \oplus U^\perp = V$
(c) Show that $(U^\perp)^\perp = U$
(d) Which of these statements remain true over a field of positive characteristic?
I was able to answer parts (a)-(c) just fine. I'm uncertain about part (d). I know that a field of characteristic is a field such that $n \bullet 1 = 0$ , for some positive integer $n$. I don't know how that would affect any of the above results. Any help on this is appreciated.