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Title: Coordinates and Basis Isomorphism
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Series: Abstract Linear Algebra
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Chapter: General vector spaces
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YouTube-Title: Abstract Linear Algebra 5 | Coordinates and Basis Isomorphism
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Bright video: https://youtu.be/v7I8ba6aAAQ
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Dark video: https://youtu.be/DeOpfSd4h_U
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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: ala05_sub_eng.srt missing
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Timestamps
00:00 Introduction
00:37 Assumptions
02:31 Definition: Coordinates with respect to a basis
03:40 Coordinate vector
03:58 Picture for the idea
06:44 Definition: basis isomorphism
08:32 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $V$ be a $3$-dimensional vector space with basis $\mathcal{B} = (b_1, b_2, b_3)$. What is correct for the basis isomorphism $\Phi_{\mathcal{B} }$?
A1: $ \Phi_{\mathcal{B} }: V \rightarrow \mathbb{F}^3 $
A2: $ \Phi_{\mathcal{B} }: V \rightarrow \mathbb{F}^2 $
A3: $ \Phi_{\mathcal{B} }: \mathbb{F}^2 \rightarrow V $
A4: $ \Phi_{\mathcal{B} }(b_1) = 1$
Q2: Let $V$ be a $2$-dimensional vector space with basis $\mathcal{B} = (b_1, b_2)$. What is correct for the basis isomorphism $\Phi_{\mathcal{B} }$?
A1: $ \Phi_{\mathcal{B} }(b_1) = \binom{1}{0} $
A2: $ \Phi_{\mathcal{B} }(b_1) = \binom{0}{1} $
A3: $ \Phi_{\mathcal{B} }(b_2) = \binom{1}{0} $
A4: $ \Phi_{\mathcal{B} }(b_2) = 2 $
Q3: Let $V$ be a $10$-dimensional vector space with basis $\mathcal{B}$. What is not a property of the basis isomorphism $\Phi_{\mathcal{B} }$?
A1: $ \Phi_{\mathcal{B} }( \lambda x ) = \lambda \Phi_{\mathcal{B} }(x)$
A2: $ \Phi_{\mathcal{B} }( x + y ) = \Phi_{\mathcal{B} }(x) + \Phi_{\mathcal{B} }(y) $
A3: $ \Phi_{\mathcal{B} }$ is linear
A4: $ \Phi_{\mathcal{B} }$ is additive
A5: $ \Phi_{\mathcal{B} }$ is a bounded map
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Last update: 2024-10