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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45ccc3c2-853e-4ee6-91f9-7e4723419012%2Fa0e519e1-1ab5-48ad-9d9e-9a7785f496da%2Fu6jv2ep_processed.png&w=3840&q=75)
Transcribed Image Text: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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