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Title: Combinations of Linear Maps
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Series: Abstract Linear Algebra
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YouTube-Title: Abstract Linear Algebra 23 | Combinations of Linear Maps
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Bright video: https://youtu.be/p0XQLJMzbq8
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Dark video: https://youtu.be/zbhyV9_gmjg
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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: ala23_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 $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a linear map given by $f(x_1, x_2) = \begin{pmatrix} 1 & 2 \ 2 & -1 \ -3 & 0 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix}$ and $g: \mathbb{R}^3 \rightarrow \mathbb{R}$ given by $g(x_1, x_2, x_3) = x_1 + x_2 + x_3$. What is the composition $ g \circ f$?
A1: It’s a linear map $g \circ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ given by $(g \circ f)(x_1, x_2) = x_2 $
A2: It’s a linear map $g \circ f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 $ given by $(g \circ f)(x_1, x_2, x_3) = \begin{pmatrix} x_1 \ x_2 \end{pmatrix} $
A3: It’s a linear map $g \circ f: \mathbb{R} \rightarrow \mathbb{R} $ given by $(g \circ f)(x) = x $
A4: It’s not a linear map.
Q2: Consider the set of linear maps between two $\mathbb{F}$-vector spaces $V$ and $W$, both of finite dimension, denoted by $\mathcal{L}(V,W)$. What is correct?
A1: $\mathcal{L}(V,W)$ is an $\mathbb{F}$-vector space of dimension $\dim(V) \cdot \dim(W)$.
A2: $\mathcal{L}(V,W)$ is not a vector space.
A3: $\mathcal{L}(V,W)$ is an $\mathbb{F}$-vector space of dimension $\dim(V) + \dim(W)$.
A4: $\mathcal{L}(V,W)$ is the empty set.
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Last update: 2024-10