00:00 Introduction

00:33 Span in the function space

01:27 Is this a basis?

02:03 Checking for linear independence

05:40 Reformulating as matrix-vector multiplication

06:52 Uniquely solvable?

08:52 Basis isomorphism

11:16 Credits

Q1: Let $V$ be the real vector space of functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Are $\cos$ and $\sin$ linearly independent?

A1: Yes!

A2: No!

A3: Only for $x = 0$.

Q2: Let $V$ be the real vector space of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $U$ the subspace spanned by the two vectors $\cos$ and $\sin$. For $\mathcal{B} = (\cos, \sin)$ let $\Phi_{\mathbb{B}}$ be the corresponding basis isomorphism. What is $\Phi_{\mathbb{B}}( 3 \sin + 7 \cos)$?

A1: $\begin{pmatrix} 7 \ 3 \end{pmatrix}$

A2: $\begin{pmatrix} 3 \ 7 \end{pmatrix}$

A3: $\begin{pmatrix} 7 \ 3 \ 0 \end{pmatrix}$

A4: $\begin{pmatrix} 0 \ 3 \ 7 \end{pmatrix}$