Suppose $ X $ and $ Y $ are rings and that $ f : X \to Y $ is an isomorphism. Let $ X_{2 \times 2} $ be the set of all $ 2 \times 2 $ matrices with components from $ X $ and $ Y_{2 \times 2} $ the set of all $ 2 \times 2 $ matrices with components from $ Y $. Show that $ X_{2 \times 2} $ and $ Y_{2 \times 2} $ are also isomorphic by completing parts below.Define $ g : X_{2 \times 2} \to Y_{2 \times 2} $ by \[g \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} f(a) & f(b) \\ f(c) & f(d) \end{bmatrix}\]Verify that $ g $ is a bijection.

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Suppose X and Y are rings and that f : X → Y is an isomorphism. Let X2x2 be the set of all 2 × 2 matrices with components from X and Y2x2 the set of all2 × 2 matrices with components from Y. Show that X2x2 and Y2x2 are also isomorphic by completing parts below. Define g: X2x2 → Y2x2 by [f(a) f(b) [f(c) f(d] [a b d Verify that g is a bijection.

Suppose X and Y are rings and that f : X → Y is an isomorphism. Let X2x2 be the set of all 2 × 2 matrices with components from X and Y2x2 the set of all2 × 2 matrices with components from Y. Show that X2x2 and Y2x2 are also isomorphic by completing parts below. Define g: X2x2 → Y2x2 by [f(a) f(b) [f(c) f(d] [a b d Verify that g is a bijection.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose $ X $ and $ Y $ are rings and that $ f : X \to Y $ is an isomorphism. Let $ X_{2 \times 2} $ be the set of all $ 2 \times 2 $ matrices with components from $ X $ and $ Y_{2 \times 2} $ the set of all $ 2 \times 2 $ matrices with components from $ Y $. Show that $ X_{2 \times 2} $ and $ Y_{2 \times 2} $ are also isomorphic by completing parts below.

Define $ g : X_{2 \times 2} \to Y_{2 \times 2} $ by

\[
g \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} f(a) & f(b) \\ f(c) & f(d) \end{bmatrix}
\]

Verify that $ g $ is a bijection.

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