• Title: Coordinates and Basis Isomorphism

  • Series: Abstract Linear Algebra

  • YouTube-Title: Abstract Linear Algebra 5 | Coordinates and Basis Isomorphism

  • Bright video: https://youtu.be/v7I8ba6aAAQ

  • Dark video: https://youtu.be/DeOpfSd4h_U

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ala05_sub_eng.srt missing

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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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