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Title: Transformation Matrix
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Series: Abstract Linear Algebra
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Chapter: General vector spaces
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YouTube-Title: Abstract Linear Algebra 8 | Transformation Matrix
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Bright video: https://youtu.be/wTHQNYhxv-M
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Dark video: https://youtu.be/CyW7Y5UFY10
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ala08_sub_eng.srt missing
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Timestamps
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
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Subtitle in English (n/a)
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Quiz Content
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: $T_{\mathcal{C} \leftarrow \mathcal{B} } = \begin{pmatrix} 2 & 4 & 0 \ 0 & 2 & 1 \ 1& 2 & 1\end{pmatrix}$
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Last update: 2024-10