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Title: Linear maps induce matrices
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Series: Linear Algebra
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Chapter: Matrices and linear systems
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YouTube-Title: Linear Algebra 20 | Linear maps induce matrices
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Bright video: https://youtu.be/9UcgdR_X2Ys
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Dark video: https://youtu.be/jZqb1RwbCCw
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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: la20_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: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map given by $$f(x,y) = \begin{pmatrix} 2y + 3 x \ 5 x - y \end{pmatrix}$$ What is the corresponding matrix $A$ such that $$f(x,y) = A \binom{x}{y}$$ holds?
A1: $$ A = \begin{pmatrix} 3 & 2 \ 5 & -1 \end{pmatrix} $$
A2: $$ A = \begin{pmatrix} 3 & -2 \ -5 & -1 \end{pmatrix} $$.
A3: $$ A = \begin{pmatrix} 2 & 3 \ 5 & -1 \end{pmatrix} $$
A4: $$ A = \begin{pmatrix} 2 & 3 \ 5 & 1 \end{pmatrix} $$
Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map. Is it possible that $f_A = f_B$ for different matrices $A,B \in \mathbb{R}^{2 \times 2}$.
A1: No, never.
A2: Yes, always.
A3: Only if $f = 0$.
Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map. What is the first column of the matrix $A$ such that $f = f_A$?
A1: $f(e_1)$
A2: $f(e_2)$
A3: $f(e_1 + e_2)$
A4: One needs more information.
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Last update: 2024-10