13. Let 4 : M₂(R) → R defined by p(d)) = a + d. Recall that the operation on M₂(R) is the usual additionCof matrices. Determine, with proof, if is a group homomorphism. If it is, find its Kernel.

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Answered: 13. Let : M₂ (R) → R defined by ( oll c…

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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11.1.2 Groups Homomorphisms
Working with Large Numbers Assessment 9/11 answered Score: 15/20 Question 1 A small country had a population of 4.8 x 10' in 1990. Since then, the original population has doubled, and an additional 8.3 x 10° people have immigrated into the country. What is the population of the country now? The population is A x 105 where A = B = Submit Question ype here to search 2 Pris F3 F4 FS F6 F7 F8 F9 F10 F11 F12 %23 & 3 4. 5 8 19. T Y U C D H J K L FL
2. Let d≥ 1, let F be a field and let Affa(F) denote the set of all functions TA, FdFd which have the form T₁,v(x) = Ax + v, where A is an invertible d x d matrix with entries in F and where v Fd. Prove that Affa(F) is a group (where the binary operation is given by composition of functions).
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.)
Also indicate which theorem or property you are using please
Let T : R² → P2(R) and U : Rª –→ M2x2 (R) be lincar transformations. A student claims T cannot be invertible because dim(R²) + dim(P2(R)). If the student is correct, prove their claim. If the student is not correct, explain why and give an example to illustrate. Clearly state whether or not the student is correct as part of your solution.
2. Suppose T: R¹4 → R¹4 is a linear transformation and the rank of T is 10. (a) Determine whether T is injective. (b) Determine whether T is surjective. (c) Determine whether T is invertible.
If T : R3 [a] → M2x2 (R) is a linear transformation then T is an isomorphism. True O False

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13. Let : M₂ (R) → R defined by ( oll c C d)) = c = a + d. Recall that the operation on M₂ (R) is the usual addition of matrices. Determine, with proof, if is a group homomorphism. If it is, find its Kernel.