15. Let 0: M2(Z) → Z where M₂(Z) is the ring of 2 × 2 matrices over the integers Z.Prove or disprove that each of the following mappings is a homomorphism.ba.o ([a= ad - bcb. 0• ([a b])= a + d (This mapping is called the trace of the matrix.)

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Answered: 15. Let 0: M2(Z) → Z where M₂(Z) is the…

Algebra and Trigonometry (6th Edition)

6th Edition

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Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Each of the following matrices determines a linear map from R into R3: 1 2 0 1 1 0 2 -1) (1) A = 2-1 2-1 (ii) B = 2 3-1 1 1 -3 2 -2 -2 0-5 3 Find a basis and the dimension of the image U and the kernel W of each map.
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Suppose that B = S-¹AS for some n x n matrices A, B and some n x n invertible matrix S. a. Show that if is in ker B, then Sa is in ker A. b. Show that the linear transformation T (*) = S from ker B to ker A is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B.
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Let T: R² R² be defined by T and C Given Pc 3 -{]-[H]} -1223) , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). [T](PB (u)) 7 [4₁]) B-{B] R]} = Let u = = Ex: 5 [-3x1 + 2x₂ / -4

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15. Let 0: M2(Z) → Z where M₂(Z) is the ring of 2 × 2 matrices over the integers Z. Prove or disprove that each of the following mappings is a homomorphism. b a. o ([a = ad - bc b. 0 • ([a b]) = a + d (This mapping is called the trace of the matrix.)