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A new cruise ship line has just launched 3 new ships. The Pacific Paradise, the Caribbean Paradise, & the Mediterranean Paradise. The Caribbean Paradise has 16 more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 40 fewer deluxe staterooms than 4 times the number of deluxe staterooms on the Pacific Paradise. Find the number of deluxe staterooms for each of the ship if the total number of deluxe staterooms for the 3 ships is 836.
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Suppose that {v1.....Vn} is a basis for the vector space V. Given any vector v element of V, we can express v as a linear combination v = X1V1+.....+XnVn. The uniqueness of this expression means that mapping v to the n-tuple of coefficients (X1...Xn) defines a function O: V - R^n. Prove that this function is a bijection.
Answered!
The no of crime committed in a city everyday from 1am to 2am has poisson distribution.the probability of no crime during this hour is 0.05.find the average number of crimes committed during this hour?
In Progress...
Give an example of a pair of finite groups G,G' such that, for some field k, kG ∼= kG' as k-algebras, but G is not isomorphic to G' as groups.
In Progress...
Let k be a field whose characteristic is prime to the order of a finite group G. Show that the following two statements are equivalent:
(a) each irreducible kG-module has k-dimension 1;
(b) G is abelian, and k is a splitting field for G.
In Progress...
Let R = kG where k is any field and G is any group. Let I be the ideal of R generated by ab − ba for all a, b ∈ R. Show that R/I ∼ k[G/G'] as k-algebras, where G' denotes the commutator subgroup of G.
In Progress...
Let R = kG where k is any field and G is any group. Let I be the ideal of R generated by ab − ba for all a, b ∈ R. Show that I = (sum over a∈G) (a − 1)kG.
In Progress...
For any field k and for any normal subgroup H of a group G, show that kH ∩ rad kG = rad kH.
In Progress...
For any field k and for any normal subgroup H of a group G, assume further that [G : H] is finite and prime to char k. Let V be a kG-module and W be a kH-module. Show that V is a semisimple kG-module iff kHV is a semisimple kH-module.
In Progress...
For any field k and for any normal subgroup H of a group G, assume further that [G : H] is finite and prime to char k. Let V be a kG-module and W be a kH-module. Show that W is a semisimple kH-module iff the induced module kG ⊗kH W is a semisimple kG-module.
In Progress...
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