• Title: Coordinates and Basis Isomorphism

• Series: Abstract Linear Algebra

• Chapter: General vector spaces

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

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

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

• Subtitle on GitHub: ala05_sub_eng.srt missing

• 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

• Subtitle in English (n/a)
• 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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