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Title: Orthogonality
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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 13 | Orthogonality
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Bright video: https://youtu.be/YCW1zTk4C9w
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Dark video: https://youtu.be/iTSbHzXhagA
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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: ala13_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 vector space with inner product $\langle \cdot, \cdot \rangle$. Which statement is not correct?
A1: If $x \perp y$, then $-x$ is not orthogonal to $y$.
A2: The zero vector orthogonal to the zero vector.
A3: The zero vector is orthogonal to each vector $x \in V$.
A4: If $x,y \in V$ are orthogonal, then $\langle x, y \rangle = 0$.
Q2: Let $V = \mathbb{C}^2$ be given with the standard inner product $\langle y, x \rangle = \overline{y_1} x_1 + \overline{y_2} x_2$. Which statement is correct?
A1: The vector $\binom{i}{1}$ is orthogonal to $\binom{i}{1}$.
A2: The vector $\binom{i}{1}$ is orthogonal to $\binom{-i}{1}$.
A3: The vector $\binom{1}{1}$ is orthogonal to $\binom{i}{i}$.
A4: The vector $\binom{0}{1}$ is not orthogonal to $\binom{1}{0}$.
Q3: Let $V = \mathcal{P}([0,1])$ be given with the standard inner product $\langle y, x \rangle = \overline{y_1} x_1 + \overline{y_2} x_2$. Which statement is correct?
A1: The vector $\binom{i}{1}$ is orthogonal to $\binom{i}{1}$.
A2: The vector $\binom{i}{1}$ is orthogonal to $\binom{-i}{1}$.
A3: The vector $\binom{1}{1}$ is orthogonal to $\binom{i}{i}$.
A4: The vector $\binom{0}{1}$ is not orthogonal to $\binom{1}{0}$.
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Last update: 2024-10