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Title: Eigenvalues and Eigenvectors for Linear Maps
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Series: Abstract Linear Algebra
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Chapter: General linear maps
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YouTube-Title: Abstract Linear Algebra 34 | Eigenvalues and Eigenvectors for Linear Maps
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Subtitle on GitHub: ala34_sub_eng.srt missing
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $\ell: V \rightarrow V$ be a linear map and $\lambda \in \mathbb{F}$ be an eigenvalue of $\ell$. What is never possible?
A1: $\mathrm{dim}\mathrm{Ker}(\ell - \lambda \mathrm{id}) = 0 $
A2: $\lambda = 0$.
A3: $\mathrm{det}(\ell - \lambda \mathrm{id}) = 0$
A4: $\mathrm{Ran}(\ell - \lambda \mathrm{id}) \subsetneq V $
Q2: Let $\ell: V \rightarrow V$ be a linear map where $V = \mathcal{P}_3(\mathbb{R})$ and $\ell(p) = p^\prime$. Is there an eigenvector to the eigenvalue $\lambda = 0$?
A1: Yes, the constant polynomial $x \mapsto 3$.
A2: Yes, the constant polynomial $x \mapsto 0$.
A3: Yes, the linear polynomial $x \mapsto x$.
A4: No, there are no eigenvectors, so $0$ is not an eigenvalue of $\ell$.
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Date of video: 2024-11-20
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Last update: 2025-10
