00:00 Introduction

00:47 Picture for change of basis

02:01 What happens with the first unit vector?

03:23 Representation by a matrix

04:24 Change-of-basis matrix

05:20 Picture for the transition matrix

06:40 Inverse of the transformation Matrix

07:10 Example with polynomials

10:38 Credits

Q1: Let $V$ be an $n$-dimensional complex vector space and $\mathcal{B}$, $\mathcal{C}$ be two bases of $V$. What is correct for the change-of-basis matrix $T_{\mathcal{C} \leftarrow \mathcal{B} }$?

A1: $T_{\mathcal{C} \leftarrow \mathcal{B} } \in \mathbb{C}^{n \times n}$

A2: $T_{\mathcal{C} \leftarrow \mathcal{B} } \in \mathbb{R}^{n \times n}$

A3: $T_{\mathcal{C} \leftarrow \mathcal{B} } \in \mathbb{R}^{n-1 \times n}$

A4: $T_{\mathcal{C} \leftarrow \mathcal{B} } \in \mathbb{C}^{n}$

Q2: Let $V$ be an $n$-dimensional complex vector space and $\mathcal{B}$, $\mathcal{C}$ be two different bases of $V$. What is correct for the change-of-basis matrix $T_{\mathcal{C} \leftarrow \mathcal{B} }$?

A1: $T_{\mathcal{C} \leftarrow \mathcal{B} }$ is invertible

A2: $T_{\mathcal{C} \leftarrow \mathcal{B} }^{-1} = T_{\mathcal{C} \leftarrow \mathcal{B} }$

A3: $T_{\mathcal{C} \leftarrow \mathcal{B} }$ is singular

A4: $T_{\mathcal{C} \leftarrow \mathcal{B} } = T_{\mathcal{B} \leftarrow \mathcal{C} } $

Q3: Let $V$ be an $3$-dimensional vector space and $\mathcal{B} = (b_1, b_2, b_3)$ be a basis of $V$. Define a new basis $\mathcal{C} = (2 b_1 + b_3, 4 b_1+ 2 b_2 + 2 b_3, b_2 + b_3)$. What is the change-of-basis matrix in this case?

A1: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 2 & 4 & 0 \ 0 & 2 & 1 \ 1& 2 & 1\end{pmatrix}^{-1}$

A2: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 1 & 2 & 3 \ 2 & 3 & 1 \ 3& 1 & 2\end{pmatrix}$

A3: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 2 & 0 & 1 \ 4 & 2 & 2 \ 0& 1 & 1\end{pmatrix}$

A4: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 2 & 0 & 1 \ 4 & 2 & 2 \ 0& 1 & 1\end{pmatrix}^{-1}$

A5: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 2 & 4 & 0 \ 0 & 2 & 1 \ 1& 2 & 1\end{pmatrix}$