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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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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: ala17_sub_eng.srt missing
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Download dark video: Link on Vimeo
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Definitions in the video: approximation formula
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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: 2025-10
