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Title: Orthonormal Basis
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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 18 | Orthonormal Basis
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Bright video: https://youtu.be/VfpfHxUF630
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Dark video: https://youtu.be/CTKOV9c8VFE
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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: ala18_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 with an ONB $\mathcal{B} = (e_1, \ldots, e_k)$. What is not correct?
A1: $\langle b_1 - b_2, b_2 \rangle = 0$
A2: $\mathcal{B}$ is a basis.
A3: $\langle b_1, b_2 \rangle = 0$
A4: $\langle b_2, b_2 \rangle = 1$
Q2: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a $k$-dimensional subspace with an ONS $\mathcal{B} = (e_1, \ldots, e_n)$. What is not correct?
A1: $\langle b_1 - b_2, b_2 \rangle = 1$
A2: $\mathcal{B}$ is a basis.
A3: $\langle b_1, b_2 \rangle = 0$
A4: $\langle b_2, b_2 \rangle = 1$
Q3: Consider $\mathbb{R}^2$ with inner product $\langle x,y\rangle = x_1 x_2 + 3 y_1 y_2$. Is $\left( \binom{1}{0}, \binom{0}{1} \right)$ an ONB?
A1: No, it’s not even a basis.
A2: No, it’s not even an orthogonal basis.
A3: No, it’s an orthogonal basis but not an ONB.
A4: Yes, it is.
A5: One needs more information.
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Last update: 2024-10