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Title: Projection Theorem
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Series: Hilbert Spaces
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Chapter: Orthogonality
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YouTube-Title: Hilbert Spaces 9 | Projection Theorem
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Quiz: Test your knowledge
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Subtitle on GitHub: hs09_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X$ be an inner product space and $U \subseteq X$ a subset. What can we say about the intersection $ U \cap U^\perp$?
A1: It’s either empty or $ U \cap U^\perp = { 0 } $.
A2: It’s empty.
A3: It always satisfies $ U \cap U^\perp = { 0 } $.
A4: It satisfies $ U \cap U^\perp \supseteq { 0 } $.
Q2: Let $X$ be an inner product space and $U \subseteq X$ be a subspace. Let $p + n = \tilde{p} + \tilde{n}$ for $p, \tilde{p} \in U$ and $n, \tilde{n} \in U^\perp$. What is always correct?
A1: $p = \tilde{p}$ and $n = \tilde{n}$.
A2: $p = 0$ and $n = 0$.
A3: $p = 0$ or $n = 0$.
A4: $p - \tilde{p} = n$
Q3: Let $X$ be an inner product space and $U \subseteq X$ be a closed subspace. For $x \in X$, let’s define $p \in U$ as the unique best approximation. What is always correct?
A1: $x - p \in U^\perp$
A2: $x - p = 0$
A3: $p = 0$
A4: $n = 0$
A5: $x + p \perp U$.
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Date of video: 2025-02-13
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Last update: 2026-02
