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Title: Example for General Linear Equation
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Series: Abstract Linear Algebra
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YouTube-Title: Abstract Linear Algebra 32 | Example for General Linear Equation
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Bright video: https://youtu.be/bkQ2L-fyLpw
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Dark video: https://youtu.be/zL1nu5TVoeY
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Quiz: Test your knowledge
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Subtitle on GitHub: ala32_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 $\ell: V \rightarrow V$ be a linear map where $V = \mathcal{P}_3(\mathbb{R})$ and $\ell(p) = p^\prime$. What is not correct?
A1: $\mathrm{dim}( \mathrm{Ran} ) (\ell) = 4$
A2: $\mathrm{dim}( \mathrm{Ran} ) (\ell) = 3$
A3: $\mathrm{dim}( \mathrm{Ker} ) (\ell) = 1$
A4: $\mathrm{dim}( \mathrm{Ker} ) (\ell) + \mathrm{dim}( \mathrm{Ran} ) (\ell) = 4$
Q2: Let $\ell: V \rightarrow V$ be a linear map where $V = \mathcal{P}3(\mathbb{R})$ and $\ell(p) = p^\prime$. Let $\mathcal{B} = (m_0, m_1, m_2, m_3)$ be a basis of $V$. What is $\ell{\mathcal{B} \leftarrow \mathcal{B}}$?
A1: $\ell_{\mathcal{B} \leftarrow \mathcal{B}} = \begin{pmatrix} 0 & 1 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 3 \ 0 & 0 & 0 & 0 \ \end{pmatrix}$
A2: $\ell_{\mathcal{B} \leftarrow \mathcal{B}} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 3 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 1 & 0 \ \end{pmatrix}$
A3: $\ell_{\mathcal{B} \leftarrow \mathcal{B}} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 3 & 0 & 0 \ \end{pmatrix}$
A4: $\ell_{\mathcal{B} \leftarrow \mathcal{B}} = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 2 & 0 & 0 \ \end{pmatrix}$
Q3: Let $\ell: V \rightarrow V$ be a linear map where $V = \mathcal{P}_3(\mathbb{R})$ and $\ell(p) = p^\prime$. Is the equation $\ell(x) = b$ solvable?
A1: Only if $b \in \mathrm{Span}(m_0, m_1, m_2)$.
A2: Only if $b \in \mathrm{Ker}(\ell)$.
A3: Only if $b \neq 0$
A4: No, never!
A5: Yes, always!
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Date of video: 2024-10-13
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Last update: 2025-01
