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Title: Column Picture of the Matrix-Vector Product
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Series: Linear Algebra
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Chapter: Matrices and linear systems
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YouTube-Title: Linear Algebra 14 | Column Picture of the Matrix-Vector Product
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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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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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: la14_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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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Which of the following claims is correct?
A1: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the columns of $A$.
A2: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^m $, the vector $A \mathbf{x}$ is well-defined.
A3: For $ A \in \mathbb{R}^{n \times m} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is well-defined.
A4: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the rows of $A$.
Q2: Which of the following equations is correct?
A1: $\displaystyle \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = $ $\displaystyle x_1 \begin{pmatrix} 2 \ 0 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 1 \end{pmatrix} $
A2: $ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = $ $\displaystyle x_1 \begin{pmatrix} 2 \ 1 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 0 \end{pmatrix} $
A3: $\displaystyle \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = $ $\displaystyle 0 $
A4: $\displaystyle \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = $ $\displaystyle \begin{pmatrix}x_1 \ x_2 \end{pmatrix} $
A5: $\displaystyle \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = $ $\displaystyle \begin{pmatrix} 2 x_1 \ x_2 \end{pmatrix} $
Q3: A matrix $A \in \mathbb{R}^{m \times n}$ defines a map $f_A$. What is the correct definition?
A1: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto A \mathbf{x}$
A2: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A \mathbf{x}$
A3: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A - \mathbf{x}$
A4: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A + \mathbf{x}$
A5: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto \mathbf{x}$
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Last update: 2024-10
