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Title: Homomorphisms and Isomorphisms
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Series: Abstract Linear Algebra
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YouTube-Title: Abstract Linear Algebra 24 | Homomorphisms and Isomorphisms
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Bright video: https://youtu.be/Xbo9nW0Mm94
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Dark video: https://youtu.be/Ov3RrWwpg_s
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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: ala24_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 $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be injective. Which is statement is always correct?
A1: If $(v_1,v_2, \ldots, v_k)$ is linearly independent in $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is linearly independent in $W$.
A2: $f$ is also surjective.
A3: $f$ is also invertible.
A4: If $\mathcal{F} = (v_1,v_2, \ldots, v_k)$ is a family with $\mathrm{Span}(\mathcal{F}) = V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ also spans $W$.
Q2: Let $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be surjective. Which is statement is always correct?
A1: If $(v_1,v_2, \ldots, v_k)$ is linearly independent in $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is linearly independent in $W$.
A2: $f$ is also injective.
A3: $f$ is also invertible.
A4: If $\mathcal{F} = (v_1,v_2, \ldots, v_k)$ is a family with $\mathrm{Span}(\mathcal{F}) = V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ also spans $W$.
Q3: Let $f: V \rightarrow W$ be a linear map between two $\mathbb{F}$-vector spaces $V$ and $W$ and let $f$ be bijective. Which is statement is not correct in general?
A1: If $(v_1,v_2, \ldots, v_k)$ is a basis of $V$, then $(f(v_1) , f(v_2), \ldots, f(v_k) )$ is a basis of $W$.
A2: $f^{-1}$ is linear.
A3: $f$ is also invertible.
A4: $f(0) = 0$
A5: $f^{-1} \circ f$ is not a linear map.
A6: $f$ is an isomorphism.
Q4: Let $f: V \rightarrow W$ be an isomorphism between two $\mathbb{F}$-vector spaces $V$ and $W$. What is always correct?
A1: $\dim(V) = \dim(W)$
A2: $\dim(V) < \dim(W)$
A3: $\dim(V) > \dim(W)$
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Last update: 2024-10