00:00 Introduction

00:36 Abstract vector space

01:38 Behaviour of the zero vector

03:00 Proof for the formulas

07:13 Linear subspaces = special vector spaces

10:00 Definition of linear subspace

12:02 Example quadratic polynomials

14:07 Credits

Q1: Let $V$ be a vector space with zero vector $\mathbf{o}$. Which claim is correct?

A1: $0 \cdot v = \mathbf{o}$ for all $v \in V$.

A2: $(-1) \cdot v = \mathbf{o}$ for all $v \in V$.

A3: $\lambda \cdot v = \mathbf{o}$ for all $v \in V$ and all $\lambda \in \mathbb{F}$.

A4: $(-1) \cdot v + v = v$ for all $v \in V$.

Q2: Let $V$ be a vector space and $U \subseteq V$ a subspace. What is not correct in general?

A1: $u + v \in U$ for each $u \in U$ and $v \in V$.

A2: $u + v \in U$ for all $u,v \in U$.

A3: $\lambda \cdot u \in U$ for all $ u \in U$ and all $\lambda \in \mathbb{F}$.

A4: The zero vector lies in $U$.