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Title: Approximation Formula
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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 17 | Approximation Formula
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Bright video: https://youtu.be/aWZVpJ0CfoA
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Dark video: https://youtu.be/zyVE7ttc5Rk
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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: ala17_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 finite-dimensional subspace. For a vector $x \in V$ we have the orthogonal projection $x = x|U + x|{U^{\perp}}$. What is the distance between $x$ and $U$?
A1: $| x|_U | $
A2: $| x|_{U^{\perp}}| $
A3: $| x | $
A4: $| x + x|_{U^{\perp}} | $
Q2: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $u,v \in V$ two vectors with $u \perp v$. What is always correct?
A1: $| u + v |^2 = | u |^2 + |v |^2$
A2: $| u + v | = | u | + |v |$
A3: $| u + v | = | u |^2$
A4: $| u + v |^2 = 0$
Q3: Let $\mathbb{R}^3$ given with the standard inner product $\langle \cdot, \cdot \rangle$ and $U$ the two-dimensional subspace given by the $x$-$y$-plane. What is the distance between $U$ and $x= \begin{pmatrix} 5 \ 4 \ 3 \end{pmatrix}$?
A1: 5
A2: 4
A3: 3
A4: 0
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Last update: 2024-10