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Title: Examples of Linear Maps
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Series: Linear Algebra
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Chapter: Matrices and linear systems
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YouTube-Title: Linear Algebra 21 | Examples of Linear Maps
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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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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: la21_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: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors that have an angle $\alpha$ between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?
A1: It’s also $\alpha$.
A2: It’s $\frac{1}{2}\alpha$.
A3: It’s $\frac{1}{3}\alpha$.
A4: It’s $-\alpha$.
A5: One needs more information.
A6: It’s $0$.
Q2: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors that have an angle $\alpha$ between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?
A1: It’s also $\alpha$.
A2: It’s $\frac{1}{2}\alpha$.
A3: It’s $\frac{1}{3}\alpha$.
A4: It’s $-\alpha$.
A5: One needs more information.
A6: It’s $0$.
Q3: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix}\cos(\alpha) & -\sin(\alpha) \ \sin(\alpha) & \cos(\alpha) \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors with a right-angle between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?
A1: It’s also a right-angle.
A2: It’s not a right-angle anymore.
A3: One needs more information.
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Last update: 2024-10
