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Title: Rank-Nullity Theorem
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Series: Linear Algebra
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Chapter: Matrices and linear systems
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YouTube-Title: Linear Algebra 35 | Rank-Nullity Theorem
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Bright video: https://youtu.be/Ia6J9uwGWgw
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Dark video: https://youtu.be/jPyxiNY7a88
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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: la35_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What is the correct definition for the rank of a matrix $A \in \mathbb{R}^{m\times n}$?
A1: $\mathrm{rank}(A) = \mathrm{dim}\Big({ A \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n } \Big) $
A2: $\mathrm{rank}(A) = \mathrm{dim}\Big( { \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^m } \Big)$
A3: $\mathrm{rank}(A) = \mathrm{dim} \Big( { \mathbf{y} \mid \mathbf{y} \in \mathbb{R}^m } \Big)$
A4: $\mathrm{rank}(A) = \mathrm{dim} \Big( { A^T \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^n } \Big)$
A5: $\mathrm{rank}(A) = \mathrm{dim} \Big( { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = 0 }\Big)$
Q2: What is the correct formulation of the rank-nullity theorem for a matrix $A \in \mathbb{R}^{m\times n}$?
A1: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = n$
A2: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = m$
A3: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) + n = 0$
A4: $\mathrm{dim} \Big( \mathrm{Ker}(A) \Big) + \mathrm{dim} \Big( \mathrm{Ran}(A) \Big) = n-m$
Q3: What is the rank of the matrix $\begin{pmatrix} 2 & 1 \ 0 & 0 \end{pmatrix}$?
A1: $1$
A2: $0$
A3: $2$
A4: $3$
Q4: Let $A \in \mathbb{R}^{4 \times 3}$ be matrix with rank $1$. What is the dimension of the kernel of $A$?
A1: $2$
A2: $0$
A3: $1$
A4: $3$
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Last update: 2024-10