Q1: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a subset. What is never correct?

A1: $U \cap U^\perp = { 0 }$

A2: $U \cap U^\perp \subseteq { 0 }$

A3: $U \cap U^\perp \supseteq { 0 }$

A4: $U^\perp = \emptyset$

Q2: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a subspace. What is always correct?

A1: $U \cap U^\perp = { 0 }$

A2: $U \cap U^\perp = V $

A3: $U \cup U^\perp = V$

A4: $U^\perp = \emptyset$

Q3: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a finite subset given by $U = { b_1, b_2, \ldots, b_k }$. What is not correct in general?

A1: $x \in U^\perp$ implies $x \perp b_j$ for all $j$.

A2: $x \perp b_j$ for all $j$ implies $x \in U^\perp$.

A3: $U^\perp = (\mathrm{Span}(U))^\perp$.

A4: $U \cap U^\perp = {0}$.