1. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel? = b, where M₂ (R) is the additive group of 2 x 2 matrices with entries in R. 4: M₂ (R) → R by ø ((ab) φ: C

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Chapter2: Second-order Linear Odes
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11.1.1. Groups Homomorphisms

**Question 1:**

Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?

\[
\phi: \mathbf{M}_2(\mathbb{R}) \to \mathbb{R} \text{ by } \phi\left(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\right) = b,
\]

where \(\mathbf{M}_2(\mathbb{R})\) is the additive group of \(2 \times 2\) matrices with entries in \(\mathbb{R}\).
Transcribed Image Text:**Question 1:** Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel? \[ \phi: \mathbf{M}_2(\mathbb{R}) \to \mathbb{R} \text{ by } \phi\left(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\right) = b, \] where \(\mathbf{M}_2(\mathbb{R})\) is the additive group of \(2 \times 2\) matrices with entries in \(\mathbb{R}\).
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