Q1: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a $k$-dimensional subspace. Which statement is always correct?

A1: There is an ONB for $U$.

A2: Each basis of $U$ is an ONB of $U$.

A3: There is a basis of $U$ with $k+1$ elements, which are mutually orthogonal.

A4: The Gram-Schmidt procedure is only applicable if $k \geq 2$.

Q2: Consider $V = \mathbb{R}^3$ together with the standard inner product. Let’s apply the Gram-Schmidt procedure to the basis $$ \left( \begin{pmatrix} 1\ 1 \ 1 \end{pmatrix}, \begin{pmatrix} 0\ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0\ 0 \ 1 \end{pmatrix} \right) $$ What is the correct approach in calculations here?

A1: You should change the order and apply Gram-Schmidt to $$ \left( \begin{pmatrix} 0\ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0\ 0 \ 1 \end{pmatrix}, \begin{pmatrix} 1\ 1 \ 1 \end{pmatrix} \right) $$ because then the first to steps are already done.

A2: You should just apply the algorithm without much thinking. So first we normalize $\begin{pmatrix} 1\ 1 \ 1 \end{pmatrix}$.

A3: You should just give up because Gram-Schmidt is impossible here.