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Title: Gramian Matrix
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Series: Abstract Linear Algebra
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Chapter: General inner products
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YouTube-Title: Abstract Linear Algebra 16 | Gramian Matrix
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Bright video: https://youtu.be/nokLWUK9dwM
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Dark video: https://youtu.be/guZzM62MTwo
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ala16_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
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.
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Last update: 2024-10