1. Let M₂ (R) be the ring of 2x2 real matrices under the usual matrix addition and matrix multiplication, and let End, (R²) be the ring of all R-linear maps T: R² → R² under the addition of maps and composition of maps. Define a map : M.(R)- End (R²) by O(A) = T₁ for all A € M₂(R) where T is the linear map T₁: R² R² defined by T₁(x) = AX for x € R² (where R² consists of the column vectors x). a) Prove that : M.(R)- End (R²) is a ring homomorphism by proving (A + B) = (A) +0(B) and (AB) = O(A)(B) for all A. B € M₂ (R). b) Prove that : M₂(R)→ End, (R) is a ring isomorphism by proving that is one-to- one and onto. - c) Do you believe that the idea of your proof also works for the proof of : M (R) → End (R") being a ring isomorphism? (Yes or No only).
1. Let M₂ (R) be the ring of 2x2 real matrices under the usual matrix addition and matrix multiplication, and let End, (R²) be the ring of all R-linear maps T: R² → R² under the addition of maps and composition of maps. Define a map : M.(R)- End (R²) by O(A) = T₁ for all A € M₂(R) where T is the linear map T₁: R² R² defined by T₁(x) = AX for x € R² (where R² consists of the column vectors x). a) Prove that : M.(R)- End (R²) is a ring homomorphism by proving (A + B) = (A) +0(B) and (AB) = O(A)(B) for all A. B € M₂ (R). b) Prove that : M₂(R)→ End, (R) is a ring isomorphism by proving that is one-to- one and onto. - c) Do you believe that the idea of your proof also works for the proof of : M (R) → End (R") being a ring isomorphism? (Yes or No only).
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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Transcribed Image Text:1. Let M₂ (R) be the ring of 2x2 real matrices under the usual matrix addition and matrix
multiplication, and let End, (R²) be the ring of all R-linear maps T: R² → R² under the
addition of maps and composition of maps.
Define a map
: M. (R)- End (R²) by O(A) = T₁ for all A € M₂(R) where T is
the linear map T₁: R² R² defined by T₂(X) = AX for x ER² (where R² consists of
the column vectors x).
a) Prove that : M₂(R)- End (R²) is a ring homomorphism by proving
D(A + B) = O(A) +0(B) and
(AB) = (A)(B) for all A. B € M₂(R).
b) Prove that : M₂(R)→ End (R²) is a ring isomorphism by proving that is one-to-
one and onto.
c) Do you believe that the idea of your proof also works for the proof of : M (R) -
End (R") being a ring isomorphism? (Yes or No only).
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