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Title: Gram-Schmidt Orthonormalization
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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 20 | Gram-Schmidt Orthonormalization
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Bright video: https://youtu.be/wH3vLzlHQjc
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Dark video: https://youtu.be/wRkRCgxLoFI
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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: ala20_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 $k$-dimensional subspace. Which statement is always correct?
A1: There is an ONB for $U$.
A2: Each basis of $U$ is an ONB of $U$.
A3: There is a basis of $U$ with $k+1$ elements, which are mutually orthogonal.
A4: The Gram-Schmidt procedure is only applicable if $k \geq 2$.
Q2: Consider $V = \mathbb{R}^3$ together with the standard inner product. Let’s apply the Gram-Schmidt procedure to the basis $$ \left( \begin{pmatrix} 1\ 1 \ 1 \end{pmatrix}, \begin{pmatrix} 0\ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0\ 0 \ 1 \end{pmatrix} \right) $$ What is the correct approach in calculations here?
A1: You should change the order and apply Gram-Schmidt to $$ \left( \begin{pmatrix} 0\ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0\ 0 \ 1 \end{pmatrix}, \begin{pmatrix} 1\ 1 \ 1 \end{pmatrix} \right) $$ because then the first to steps are already done.
A2: You should just apply the algorithm without much thinking. So first we normalize $\begin{pmatrix} 1\ 1 \ 1 \end{pmatrix}$.
A3: You should just give up because Gram-Schmidt is impossible here.
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Last update: 2024-10