Q1: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a subspace with basis $\mathcal{B} = (b_1, b_2, \ldots, b_k)$. What is always correct for the Gramian matrix $G(\mathcal{B})$?

A1: $G(\mathcal{B})$ is a $(k \times k)$-matrix.

A2: $G(\mathcal{B})$ has only real entries.

A3: $G(\mathcal{B})$ is singular.

A4: $G(\mathcal{B})$ is not a selfadjoint matrix.

Q2: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a subspace with basis $\mathcal{B} = (b_1, b_2, \ldots, b_k)$. Is the Gramian matrix $G(\mathcal{B})$ invertible?

A1: Yes, always!

A2: Only in the case that $\mathcal{B}$ is an ONB.

A3: Only if $\mathbb{F} = \mathbb{R}$.

A4: No, never!

Q3: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a subspace with basis $\mathcal{B} = (b_1, b_2, \ldots, b_k)$. The Gramian matrix $G(\mathcal{B})$ can be used the calculate the orthogonal projection of $x \in V$ onto $U$. How do we do that?

A1: Calculate $G(\mathcal{B})^{-1} \begin{pmatrix} \langle b_1, x \rangle \ \vdots \ \langle b_k, x \rangle \end{pmatrix}$ to get the coordinate vector of the orthogonal projection.

A2: Calculate $G(\mathcal{B}) \begin{pmatrix} \langle b_1, x \rangle \ \vdots \ \langle b_k, x \rangle \end{pmatrix}$ to get the coordinate vector of the normal component.

A3: Solve the system with $G(\mathcal{B})$ on the left-hand side and $\begin{pmatrix} \langle b_1, x \rangle \ \vdots \ \langle b_k, x \rangle \end{pmatrix}$ on the right-hand side to get the coordinates of the normal component.

A4: Calculate $G(\mathcal{B}) \begin{pmatrix} \langle b_1, b_1 \rangle \ \vdots \ \langle b_k, b_k \rangle \end{pmatrix}$ to get the coordinates of the orthogonal projection.