00:00 Introduction

00:44 Two bases for a vector space

02:53 Change of basis

03:20 Basis isomorphism properties

04:13 Example (change of basis in polynomial space)

07:35 Switching from old to new coordinates

09:35 Change of basis is a linear map

10:57 Credits

Q1: Let $V = \mathcal{P}2$ be the vector space of polynomials with degree less or equal than $2$ and $\mathcal{B} = (m_0, m_1, m_2)$ be the standard monomial basis. What is $\Phi{\mathcal{B}}(p)$ if $p$ is the polynomial $p(x) = x^2 - x^1 +8$?

A1: $\begin{pmatrix} 8 \ -1 \ 1 \end{pmatrix}$

A2: $\begin{pmatrix} 1 \ -1 \ 8 \end{pmatrix}$

A3: $\begin{pmatrix} -1 \ 1 \ 8 \end{pmatrix}$

A4: $\begin{pmatrix} 8 \ 1 \end{pmatrix}$

Q2: Let $V = \mathcal{P}_2$ be the real vector space of polynomials with degree less or equal than $2$ and $\mathcal{B} = (m_0, m_1, m_2)$ be the standard monomial basis. Moreover, let $p(x) = x^2 - x^1 +8$ be a polynomial and $\mathcal{C} = (m_0, m_1, p)$ be another basis. What is correct for the polynomial $q(x) = x^2$

A1: $\Phi_{\mathcal{B}}(q) = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$ and $\Phi_{\mathcal{C}}(q) = \begin{pmatrix} -8 \ 1 \ 1 \end{pmatrix}$

A2: $\Phi_{\mathcal{B}}(q) = \begin{pmatrix} 8 \ -1 \ 1 \end{pmatrix}$ and $\Phi_{\mathcal{C}}(q) = \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}$

A3: $\Phi_{\mathcal{B}}(q) = \begin{pmatrix} 8 \ 1 \ -1 \end{pmatrix}$ and $\Phi_{\mathcal{C}}(q) = \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}$

A4: $\Phi_{\mathcal{B}}(q) = \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}$ and $\Phi_{\mathcal{C}}(q) = \begin{pmatrix} 8 \ -1 \ 1 \end{pmatrix}$

Q3: Let $V$ be an $n$-dimensional vector space and $\mathcal{B}$ and $\mathcal{C}$ be two bases in $V$. How is the change of basis map $f$ for going from $\mathcal{B}$ to the new basis $\mathcal{C}$ defined?

A1: $f: \mathbb{F}^n \rightarrow \mathbb{F}^n$ given by $f(x) = \Phi_{\mathcal{C}} \Phi_{\mathcal{B}}^{-1}(x)$

A2: $f: \mathbb{F}^n \rightarrow \mathbb{F}^n$ given by $f(x) = \Phi_{\mathcal{B}} \Phi_{\mathcal{C}}^{-1}(x)$

A3: $f: \mathbb{F}^n \rightarrow \mathbb{F}$ given by $f(x) = \Phi_{\mathcal{B}} \Phi_{\mathcal{B}}^{-1}(x)$

A4: $f: \mathbb{F} \rightarrow \mathbb{F}^n$ given by $f(x) = \Phi_{\mathcal{B}} \Phi_{\mathcal{C}}^{-1}(x)$