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Title: Linear Maps
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Series: Abstract Linear Algebra
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Chapter: General linear maps
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YouTube-Title: Abstract Linear Algebra 22 | Linear Maps
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Bright video: https://youtu.be/I7-qk3AvEws
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Dark video: https://youtu.be/fUF4tUNYTS4
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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: ala22_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 $V,W$ be $\mathbb{F}$-vector spaces. What is always correct for a linear map $f: V \rightarrow W$?
A1: $ f(0) = 0 $
A2: $ f(x+y) = f(x-y) $
A3: $ f(\lambda \cdot x) = f(x) $
A4: $ f(0\cdot x) = f(x) $
Q3: Let $V = \mathcal{P}(\mathbb{R})$ be the space of real polynomials. Let’s define the map $\ell: V \rightarrow \mathbb{R}$ by $\ell(p) = \int_0^1 p(x) , dx$. Is it linear?
A1: No, because $\ell(0) \neq 0$.
A2: No, becasue $\ell(p+q) \neq \ell(p) + \ell(q)$ in general.
A3: No, because $\ell(x \mapsto x^2) \neq \ell(x \mapsto x) \cdot \ell(x \mapsto x)$
A4: Yes, the two properties are satisfied.
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Last update: 2024-10