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Title: Orthogonal Projection Onto Line
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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 14 | Orthogonal Projection Onto Line
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Bright video: https://youtu.be/m9_Nfzw0-Dw
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Dark video: https://youtu.be/dwkAyFuIAek
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Ad-free video: Watch Vimeo video
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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: ala14_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 $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. What is the orthogonal projection of $x \in V$ onto $\mathrm{Span}(e)$?
A1: $p = e \langle e, x \rangle$
A2: $p = \langle e, e \rangle x$
A3: $p = x \langle e, x \rangle$
A4: $p = e \langle x, x \rangle$
Q2: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. What is (in general) not a property of the map $P: V \rightarrow V$ with $ x \mapsto e \langle e, x \rangle$?
A1: $x - Px = 0$ for all $x \in V$.
A2: $P \circ P = P$
A3: $P$ is linear
A4: $\langle y, Px \rangle = \langle P y, x \rangle$ for all $x,y \in V$
A5: $\mathrm{Ran}(P) \subseteq \mathrm{Span}(e)$
Q3: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$ and $e \in V$ a vector with $\langle e,e \rangle = 1$. For $x \in V$, we can write $ x = p + n$ with $p \in \mathrm{Span}(e)$. What is (in general) not correct for $n$?
A1: $n \perp x$
A2: $n$ is called the normal component of $x$ w.r.t. $\mathrm{Span}(e)$.
A3: $n \in \mathrm{Span}(e)^\perp$
A4: $n \perp e$
A5: $n \in V$
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Last update: 2024-11
