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 with an ONB $\mathcal{B} = (e_1, \ldots, e_k)$. What is not correct?

A1: $\langle b_1 - b_2, b_2 \rangle = 0$

A2: $\mathcal{B}$ is a basis.

A3: $\langle b_1, b_2 \rangle = 0$

A4: $\langle b_2, b_2 \rangle = 1$

Q2: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a $k$-dimensional subspace with an ONS $\mathcal{B} = (e_1, \ldots, e_n)$. What is not correct?

A1: $\langle b_1 - b_2, b_2 \rangle = 1$

A2: $\mathcal{B}$ is a basis.

A3: $\langle b_1, b_2 \rangle = 0$

A4: $\langle b_2, b_2 \rangle = 1$

Q3: Consider $\mathbb{R}^2$ with inner product $\langle x,y\rangle = x_1 x_2 + 3 y_1 y_2$. Is $\left( \binom{1}{0}, \binom{0}{1} \right)$ an ONB?

A1: No, it’s not even a basis.

A2: No, it’s not even an orthogonal basis.

A3: No, it’s an orthogonal basis but not an ONB.

A4: Yes, it is.

A5: One needs more information.