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Title: Range and Kernel of a Matrix
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Series: Linear Algebra
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Chapter: Matrices and linear systems
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YouTube-Title: Linear Algebra 34 | Range and Kernel of a Matrix
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Forum: Ask a question in Mattermost
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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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Subtitle on GitHub: la34_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Definitions in the video: range of matrix, kernel of matrix
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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 range of a matrix $A \in \mathbb{R}^{m\times n}$?
A1: $\mathrm{Ran}(A) = { A \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n }$
A2: $\mathrm{Ran}(A) = { \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^m }$
A3: $\mathrm{Ran}(A) = { \mathbf{y} \mid \mathbf{y} \in \mathbb{R}^m }$
A4: $\mathrm{Ran}(A) = { A^T \mathbf{x} \mid A \mathbf{x} \in \mathbb{R}^n }$
A5: $\mathrm{Ran}(A) = { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = 0 }$
Q2: What is the correct definition for the kernel of a matrix $A \in \mathbb{R}^{m\times n}$?
A1: $\mathrm{Ker}(A) = { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = 0 }$
A2: $\mathrm{Ker}(A) = { \mathbf{x} \in \mathbb{R}^n \mid A \mathbf{x} = \mathbf{y} }$
A3: $\mathrm{Ker}(A) = { A^{-1} \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n }$
A4: $\mathrm{Ker}(A) = { A \mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n }$
Q3: What is not the range of the matrix $\begin{pmatrix} 2 & 1 \ 0 & 0 \end{pmatrix}$
A1: $\mathrm{Ran}(A) = \mathrm{Span}( \begin{pmatrix} 2 \ 1 \end{pmatrix})$
A2: $\mathrm{Ran}(A) = \mathrm{Span}( \mathbf{e}_1 )$
A3: $\mathrm{Ran}(A) = \mathrm{Span}( \begin{pmatrix} 1 \ 0 \end{pmatrix} )$
A4: $\mathrm{Ran}(A) = \mathrm{Span}(\begin{pmatrix} 2 \ 0 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \end{pmatrix} )$
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Last update: 2024-10
