$a)$ Let $\hat{A} = |e_{1}\rangle \langle e_{3}| + |e_{3}\rangle \langle e_{1}|$. Consider the space $\mathbb{R}^3$, what does $\hat{A}$ mean?
$b)$ Let $\hat{B} = |e_{1}\rangle \langle e_{1}| + |e_{3}\rangle \langle e_{3}|$. Consider the space $\mathbb{R}^3$, what does $\hat{B}$ mean?
$c)$ Let $\hat{C} = |e_{1}\rangle \langle e_{1}| + |e_{2}\rangle \langle e_{2}| + |e_{3}\rangle \langle e_{3}|$. Consider the space $\mathbb{R}^3$, what does $\hat{C}$ mean?
$d)$ Let $\hat{D} = 2|e_{1}\rangle \langle e_{1}| + 2|e_{2}\rangle \langle e_{2}| + 2|e_{3}\rangle \langle e_{3}|$. Consider the space $\mathbb{R}^3$, what does $\hat{D}$ mean?
a) $\hat A$ is the operator that interchanges the 1st and the 3rd components of any vector and annihilates its second component
b) $\hat B$ is the operator that projects any vector on the plane spanned by $|e_{1}\rangle$ and $|e_{3}\rangle$
c) $\hat C$ s the identity operator
d) $\hat D$ the operator that scales every vector by 2