00:00 Introduction

00:36 Definition of polynomial spaces

02:27 Definition: linear combination

03:38 Definition: span of a subspace

04:44 Definition: generating set for a subspace

05:29 Definition: linear independent sets

06:36 Definition: basis of a subspace

07:57 Definition: dimension of a subspace

09:18 Examples for polynomial spaces

11:25 Example for infinite-dimensional vector space

11:44 Example for matrices

12:42 Credits

Q1: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is a linear combination of vectors of $V$?

A1: $x^2 + 5 x^7 - \frac{1}{10} x^{2023}$

A2: $\sum_{j=1}^\infty x^j$

A3: $7 x^2 - \sin(x)$

A4: $x^3 - 6 \frac{1}{x}$

Q2: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is the span of the set $M$ given by the polynomials $x^2$ and $x^3$?

A1: $\mathrm{Span}(M) = { \alpha x^2 + \beta x^3 \mid \alpha, \beta \in \mathbb{R} }$

A2: $\mathrm{Span}(M) = { 0 }$

A3: $\mathrm{Span}(M) = { x^2 , x^3 }$

A4: $\mathrm{Span}(M) = { x^2 + x^3 }$

Q3: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is the dimension of the subspace $ { \alpha x + \beta x^2 + \gamma x \mid \alpha, \beta, \gamma }$?

A1: 2

A2: 3

A3: 1

A4: 0