Eigenvalue and eigenspace problem 
Let $S=\{1,2, \cdots,n\}$ be a non-empty set and $P(S)$ be the power
  set of $S$. In this set up $P(S)$ is a vector space over the field
  $\mathbb{F}_2$, and the vector addition is given by symmetric
  difference of sets, that is for $A_1,A_2 \in P(S)$
$$A_1+A_2=(A_1-A_2) \cup (A_2-A_1)$$
Now fixed an element $B \in P(S)$ and define a liner transformation
  $f:P(S) \to P(S)$ given by
$$f(A)=A\cap B$$
Questions:
1) Find the eigenvalue and eigenspace of $f$
2) If $B=\{1,2, \cdots ,m\}$, find the minimal polynomial,
  characteristic polynomial and the Jordan normal form of $f$.

Facts (previous questions):
a) $\emptyset$ is the zero element of $P(S)$ and the addition inverse of $A$ is itself.
b) $P(S)$ has an basis $\{\{1 \},\{2 \},\cdots,\{ n\} \}$, hence $\dim(P(S))=n$
c) $\ker(f)=P(S-B)$
d) $\text{Im}(f)=P(B)$
If I can write down a matrix for $f$ than I know how to do these 2 questions, however I don't know how to fine such matrix. So, I just try to write down something and see how's it goes.
Let $\lambda$ and $V$ be the eigenvalue and the corresponding eigenvector of $f$ and $E_{\lambda}$ be the eigenspace of the corresponding $\lambda$, so now we have
$$  f(V)=V \cap B=\lambda V$$
as $0 \cdot V =\emptyset$, $1 \cdot V=V$, so we have that $\lambda V=V$ or $\emptyset$
This gives $\lambda=1$ or $0$
To find $E_1$, I look at $f(V)=V$ and intuition tells me that $E_1=\text{Im}(f)$. Same for $E_0$, when I look at $f(V)=\emptyset$ I feel like it is equal to $\ker(f)$.
For question 2) I remember that minimal polynomial has something to do with eigenvalues, so if $m(X)$ is the minimal polynomial, it should of the form 
$$m(X)=X^a (X-1)^b$$
Any help would be thankful.
 A: HINT
In general, if $\mathbb F$ is a field and $\mathbb V$ is an $n$ dimensional  vector space over that field then $\mathbb V$ can be represented as vectors
$$\begin{bmatrix}
x_1\\
x_2\\
\vdots\\
x_n
\end{bmatrix}, \ x_i\in \mathbb F, \ i=1,2,...,n.$$
In our case $\mathbb F_2=\{0,1\}$ with the addition and multiplication tables are:
$$\begin{matrix}
&&\color{red}+&&&&&\color{blue}{\cdot}\\
&0&&1&&&0&&1\\
0&\color{red}0 &&\color{red}1&&&\color{blue}0&&\color{blue}0\\
1&\color{red}1&&\color{red}0&&&\color{blue}0&&\color{blue}1
\end{matrix}  $$ 
We have to show the isomorphism between the set representation and the vector representation of $P(S)$. 
The basis given by the OP. Every set is represented by a vector that contains a $1$ at the element that belongs to the set and a zero at the element that does not belong to the set.
Multiplying by $1$ does not change the vector and multiplying by $0$ chages the vector to the one representing the empty set (full of zeros.)
Vector addition corresponds to the symmetric difference. For example, let $n=3$ and take the two sets and the corresponding two vectors:
$$A=\{1\} \text{ and }, B=\{1,3\}\ ; v_A=\left[\begin{smallmatrix}1\\0\\0\end{smallmatrix}\right],v_B=\left[\begin{smallmatrix}1\\0\\1\end{smallmatrix}\right].$$
The addition in the language of sets
$$A+B=A\cap\overline B \cup B\cap \overline A=\{1\}\cap\{2\}\cup\{1,3\}\cup\{2,3\}=\{3\}.$$
The same in vector notation:
$$v_A+v_B=\begin{bmatrix}1+1\\0+0\\0+1\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}=v_{\{3\}}.$$
The question is, now, the form of the matrix belonging to the set operation $$f_B=A\cap B.$$
If for instance 
$$B=\{2,3\}$$
then
$$f_B=\begin{bmatrix}
0&0&0\\
0&1&0\\
0&0&1
\end{bmatrix}.$$Indeed, for $A=\{1,3\}$; $v_A=\left[\begin{smallmatrix}1\\0\\1
\end{smallmatrix}\right]$
$$A\cap B=\{3\}\text{ and } f_Bv_A=\begin{bmatrix}
0&0&0\\
0&1&0\\
0&0&1
\end{bmatrix}\begin{bmatrix}1\\0\\1
\end{bmatrix}=\begin{bmatrix}0\\0\\1
\end{bmatrix}=v_{\{3\}}.$$
The eigenvector of $f_B$ then is $v_B$ with the eigenvalue $1$.
I just hope that the isomorphism shown helps to answer the remaining questions.
