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prizzyjade
 one year ago
If G is a simple graph, the vector space of g is the vector space over the field Z2 of integers modulo 2, whose elements are subsets of E (G). The sum is E+F of two subsets E and F is the set odd edges in E or F but not both, and scalar multiplication is defined by1×E=E and 0xE=∅. Show that this defines a vector space over Z2 and find the basis of it.
prizzyjade
 one year ago
If G is a simple graph, the vector space of g is the vector space over the field Z2 of integers modulo 2, whose elements are subsets of E (G). The sum is E+F of two subsets E and F is the set odd edges in E or F but not both, and scalar multiplication is defined by1×E=E and 0xE=∅. Show that this defines a vector space over Z2 and find the basis of it.

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, we see that addition is commutative, and that both addition and scalar multiplication are clearly closed. show that scalar multiplication over addition distributes, i.e. k(E + F) = kE + kF. well, this is easy  0 = 0(E + F) = 0E + 0F = ∅. similarly, E + F = 1(E + F) = 1E + 1F = E + F. we get from this a free additive identity ∅ and a free scalar multiplicative identity 1. since 0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 0 are an exhaustive list of addition in Z/2Z, distribution of scalar sums over scalar multiplication is easy to demonstrate. show associativity, which is probably the hardest (and yet still overwhelmingly easy) to do

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0since addition is defined as a symmetric difference (i.e. xor, https://en.wikipedia.org/wiki/Symmetric_difference) we see that the sets of singletons consisting of individual edges (i.e. {E_1}, {E_2}, {E_3}, etc.) are a very nice basis to use here, since we can construct every other set of edges possible using these and symmetric difference

prizzyjade
 one year ago
Best ResponseYou've already chosen the best response.0please can you elaborate it further ?

prizzyjade
 one year ago
Best ResponseYou've already chosen the best response.0that's the last question i need to answer to finish my problem set in graph theory

prizzyjade
 one year ago
Best ResponseYou've already chosen the best response.0it has something to do with linear algebra right ?
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