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Title: Matrix Representations for Compositions
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Series: Abstract Linear Algebra
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YouTube-Title: Abstract Linear Algebra 26 | Matrix Representations for Compositions
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Bright video: https://youtu.be/YD92kR-Xqz8
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Dark video: https://youtu.be/QGfwW008hM4
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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: ala26_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 $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces, and $\ell: V \rightarrow W$, $k: W \rightarrow V$ linear maps. What is always correct about a matrix representation of $k \circ \ell$?
A1: It’s a square matrix.
A2: It’s an invertible matrix.
A3: It’s an orthogonal matrix.
A4: It’s a unitary matrix.
Q2: Let $V,W$ be two $\mathbb{F}$-vector spaces of dimension $2$, and $\ell: V \rightarrow W$, $k: W \rightarrow V$ linear maps, with matrix representations $\begin{pmatrix} 2 & 1 \ 3 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$, respectively. What is always correct about a matrix representation of $k \circ \ell$?
A1: It’s given by $\begin{pmatrix} 2 & 1 \ 3 & 1 \end{pmatrix}$.
A2: It’s given by $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$.
A3: It’s given by $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$.
A4: It’s an invertible matrix.
Q3: Let $V$ be an $\mathbb{F}$-vector spaces of dimension $2$ and $\ell: V \rightarrow V$ be a linear map with matrix representation $\begin{pmatrix} 2 & 5 \ 1 & 2 \end{pmatrix}$. What is a matrix representation of $ \ell^{-1} $?
A1: $\begin{pmatrix} 2 & 1 \ 2 & 1 \end{pmatrix}$.
A2: $\begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}$.
A3: $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$.
A4: Every $2\times2$-matrix can be a valid matrix representation of $ \ell^{-1} $.
A5: $\begin{pmatrix} -2 & 5 \ 1 & -2 \end{pmatrix}$.
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Last update: 2024-10