00:00 Introduction

00:36 Definition of a vector space

01:39 Examples

02:48 Function Spaces

08:01 Polynomial Space

10:00 Linear Subspace

11:55 Credits

Q1: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. What is the vector addition of $p_1(x) = x^2$ and $p_2(x) = -(x-1)^2$?

A1: $(p_1 + p_2)(x) = 2x -1$

A2: $(p_1 + p_2)(x) = 2x + 1$

A3: $(p_1 + p_2)(x) = x^2 + 2x -1$

A4: $(p_1 + p_2)(x) = x^2 + 2x + 1$

A5: It’s not defined.

Q2: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. Is the element $x \mapsto \sum_{k = 1}^n \frac{x^k}{k!}$ an element of $\mathcal{P}(\mathbb{R})$ for any given $n \in \mathbb{N}$?

A1: Yes, it is.

A2: No, it is not a polynomial.

A3: No, it’s not a well-defined function.

A4: One needs more information.

Q3: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. Is the element $x \mapsto \sum_{k = 1}^\infty \frac{x^k}{k!}$ an element of $\mathcal{P}(\mathbb{R})$?

A1: Yes, it is.

A2: No, it is not a polynomial.

A3: No, it’s not a well-defined function.

A4: One needs more information.