Understanding first part of a vector space proof The following lemma is Lemma 8.2 from the book "The Development of Number Fields.''
$\textbf{Lemma 8.2}$: Let $k, r$ be non-negative integers, and let $E$ be a $k$-dimensional
$\mathbb{F}_2$-vector space. Then the probability that $k+r$ elements that are independently
drawn from $E$, with the uniform distribution, form a spanning set for $E$ is at
least $1-2^{-r}$.
The proof begins by stating that if $H$ is a hyperplane (through the origin), then the probability that $k+r$ elements will all lie in H is $(1/2)^{k+r}$.
My questions are as follows:
(1) The way I am seeing this is that the author is claiming there is a 1/2 chance a single vector chosen lies in $H$, but I am not seeing how he is coming to that conclusion.
(2) Also, I noticed in other parts of the proof (I have omitted) that the author implicitly claims no vector could lie in two different hyperplanes at the same time, but isn't it true that hyperplanes intersect at some places?
 A: Let me give a stab at (1). Since the choice of vector $v\in E$ is uniform, then 
$$ P(v\in H) = \frac{\#\{\text{vectors in }H\}}{\#\{\text{vectors in E}\}}=\frac{2^{k-1}}{2^k} = \frac{1}{2}.$$
As for the second part, I'd have to see the argument because that assumption would be wrong. For example, if $E=\mathbb{F}_2^4$, then $v=(1,1,1,1)$ lies in both $$H_1=\{(a,b,c,d): a+b=0\},\text{ and }H_2=\{(a,b,c,d): c+d=0\},$$ which are clearly two distinct hyperplanes.
A: Fixing a basis for $E$, we can represent elements of $E$ as elements of $F_2^k$.

Then any hyperplane $H$ (not necessarily through the origin) has an equation of the form
$$a_1x_1+\cdots+a_kx_k=b$$
for some $a_1,...,a_k\in F_2$, not all zero, and some $b\in F_2$.

Note that each $a_i$ is either $0$ or $1$, and $b$ is either $0$ or $1$.

$H$ goes through the origin if and only if $b=0$, but for the discussion below, we don't need that restriction.

Let $j$ be an index such that $a_j=1$.

For $i\in\{1,...,k\}$, choose $x_i$ independently and uniformly at random from $\{0,1\}$, and let
$$s=\sum_{i\ne j}a_ix_i$$
If $s=0$, the probability that $(x_1,...,x_k)\in H$ is the probability that $x_j=b$, which is $\frac{1}{2}$.

If $s=1$, the probability that $(x_1,...,x_k)\in H$ is the probability that $x_j=b-1$, which is $\frac{1}{2}$.

Either way, the probability $(x_1,...,x_k)\in H$ is $\frac{1}{2}$.

So that answers your first question.

For your second question, let $x=(1,0,...,0)$.


*

* Assuming $k\ge 2$, $x$ lies in more than one hyperplane. For example, $x$ lies in the hyperplane with equation $x_1=1$, and it also lies in the hyperplane with equation $x_2=0$.

* Assuming $k\ge 3$, $x$ lies in more than one hyperplane through the origin. For example, $x$ lies in the hyperplane with equation $x_2=0$, and it also lies in the hyperplane with equation $x_3=0$.

