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Title: Direct Sum of Subspaces
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Series: Abstract Linear Algebra
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Chapter: Some matrix decompositions
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YouTube-Title: Abstract Linear Algebra 39 | Direct Sum of Subspaces
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: ala39_sub_eng.srt missing
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Definitions in the video: direct sum of subspaces
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $V$ be complex vector space and $U_1$ and $U_2$ be two subspaces. Is $U_1 \cup U_2$ a subspace again?
A1: Only in special cases.
A2: Yes, always.
A3: No, never!
Q2: Let $V$ be complex vector space and $U_1$ and $U_2$ be two subspaces. Is $U_1 + U_2$ a subspace again?
A1: Only in special cases.
A2: Yes, always.
A3: No, never!
Q3: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda$ be an eigenvalue. What is always correct?
A1: $\mathbb{C}^n = $ $\mathrm{Ker}((A-\lambda \mathbf{1})^k) $ $\oplus $ $ \mathrm{Ran}((A-\lambda \mathbf{1})^k)$ for every $k$ equal to the Fitting index.
A2: $\mathbb{C}^n = $ $ \mathrm{Ker}((A-\lambda \mathbf{1})^k) $ $ \oplus $ $ \mathrm{Ran}((A-\lambda \mathbf{1})^k)$ for every $k \in \mathbb{N}$.
A3: $\mathbb{C}^n = $ $ \mathrm{Ker}(A-\lambda \mathbf{1}) $ $ \oplus $ $ \mathrm{Ran}(A-\lambda \mathbf{1})$.
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Date of video: 2025-02-01
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Last update: 2025-10
