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Title: Orthogonal Complement
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Series: Hilbert Spaces
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Chapter: Orthogonality
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YouTube-Title: Hilbert Spaces 6 | Orthogonal Complement
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: hs06_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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Definitions in the video: orthogonal complement, orthogonality
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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 a Hilbert space and $x,y \in X$ linearly dependent. What is a correct implication?
A1: If $x,y$ are orthogonal, then at least one of the two vectors is the zero vector.
A2: If $x,y$ are orthogonal, then both vectors are given by the zero vector.
A3: If $x,y$ are orthogonal, then $x = y$.
A4: If $x,y$ are orthogonal, then $\langle x, y \rangle \neq 0$.
Q2: Let $X$ be a Hilbert space and $A,B \subseteq X$ be two subsets. What does it mean that $A$ is orthogonal to $B$?
A1: $\langle x, y \rangle = 0$ for all $x \in A$ and $y \in B$.
A2: $\langle x, y \rangle \neq 0$ for all $x \in A$ and $y \in B$.
A3: $\langle x, y \rangle = 1$ for all $x \in A$ and $y \in B$.
A4: $\langle x, y \rangle = 1$ for all $x \in A$ and and one $y \in B$.
Q3: Let $X$ be a Hilbert space and $B \subseteq X$ be a subset. What is in general not correct for the orthogonal complement $B^\perp$?
A1: $B^\perp$ is an open subspace in $X$.
A2: $B^\perp$ is a subspace of $X$.
A3: $B^\perp$ is a closed set in $X$.
A4: $B^\perp = (\overline{\mathrm{Span}(B)})^\perp$.
A5: $\overline{B^\perp} = (\overline{B})^\perp$.
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Date of video: 2024-10-17
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Last update: 2026-02
