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

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