Answered: Mark each of the following assertions…

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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Mark each of the following assertions True (T) or False (F). 1. The sets R and RR have the same cardinality. 2. The function q(A) = det(A) on M2(R) is a quadratic form of signature (3, 1). 3. For any two nonzero matrices A,B ∈ Mn(C), there ex- ists a λ ∈ C such that det(A + λB) = 0. 4. Every linear map T : R9 → R9 has an eigenvector. 5. Every linear map T : Rn → Rn can be expressed in the form T = DS, where D is diagonal and S ∈ On(R). 6. Let G ⊂ Isom(R2) be a discrete group such that only the identity element has a fixed-point. Then G is a group of translations. 7. If A, B ∈ GLn (R) are invertible matrices such that tr(An) = tr(Bn) for all n ∈ Z, then A is similar to B. 8. Every trilinear form is the sum of a symmetric and an antisymmetric form (i.e. ⊗3V ∼= Sym3 V ⊕ ∧3V ). 9. Any subgroup of index 5 in a group of order 5005 must be normal. 10. Every abelian subgroup of the symmetries of a dodeca- hedron is cyclic.

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