-
Title: Change of Basis for Linear Maps
-
Series: Abstract Linear Algebra
-
YouTube-Title: Abstract Linear Algebra 27 | Change of Basis for Linear Maps
-
Bright video: https://youtu.be/GLnb9LaWJYg
-
Dark video: https://youtu.be/XyIKlh5h14I
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ala27_sub_eng.srt missing
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces and $\ell: V \rightarrow W$ be a linear map. Is it possible that two different matrices represent $\ell$?
A1: Yes, but for some linear maps there is only one matrix representation.
A2: Yes, one can always find two different maps when one chooses different bases on the vector spaces.
A3: No, it’s never possible.
A4: No, this only works if $V=W$.
Q2: Let $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces and $\ell: V \rightarrow W$ be a linear map. We denote the change-of-basis matrices by $T_{\mathcal{B} \leftarrow \widetilde{\mathcal{B}}}$ and $T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{C}}$ for bases $\mathcal{B}, \widetilde{\mathcal{B}}, \widetilde{\mathcal{C}}, \mathcal{C}$. What is correct?
A1: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{B}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{C} \leftarrow \widetilde{\mathcal{C}}} $
A2: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \widetilde{\mathcal{C}}} $
A3: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \mathcal{C}} $
A4: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\mathcal{C} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \mathcal{C}} $
-
Last update: 2024-10