Q1: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. What is the orthogonal projection of $x \in V$ onto $\mathrm{Span}(e)$?

A1: $p = e \langle e, x \rangle$

A2: $p = \langle e, e \rangle x$

A3: $p = x \langle e, x \rangle$

A4: $p = e \langle x, x \rangle$

Q2: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. What is (in general) not a property of the map $P: V \rightarrow V$ with $ x \mapsto e \langle e, x \rangle$?

A1: $x - Px = 0$ for all $x \in V$.

A2: $P \circ P = P$

A3: $P$ is linear

A4: $\langle y, Px \rangle = \langle P y, x \rangle$ for all $x,y \in V$

A5: $\mathrm{Ran}(P) \subseteq \mathrm{Span}(e)$

Q3: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. For $x \in V$, we can write $ x = p + n$ with $p \in \mathrm{Span}(e)$. What is (in general) not correct for $n$?

A1: $n \perp x$

A2: $n$ is called the normal component of $x$ w.r.t. $\mathrm{Span}(e)$.

A3: $n \in \mathrm{Span}(e)^\perp$

A4: $n \perp e$

A5: $n \in V$