Cramster has the solutions to most of the questions in my book EXCEPT this one,
ID: 1941745 • Letter: C
Question
Cramster has the solutions to most of the questions in my book EXCEPT this one, so I have to post it.Determine whether the given set, together with the specified operationso f addition and scalar multiplication, is a vector space over the indicated Z(sub)p. If it is not, list all of the axioms that fail to hold.
The set of all vectors in Z(sub)2(high)n with an odd number of 1s over Z(sub)2, with the usual vector addition and scalar multiplication.
I haven't a clue what 1s is and I know Z has something to do with remainders, but don't know how to begin processing this question. I need a reply that shows how each Axion(1-10) are either satisfied or not satisfied.
Thank you!!!
Explanation / Answer
Axiom Signification Associativity of addition v1 + (v2 + v3) = (v1 + v2) + v3. Commutativity of addition v1 + v2 = v2 + v1. Identity element of addition There exists an element 0 ? V, called the zero vector, such that v + 0 = v for all v ? V. Inverse elements of addition For all v ? V, there exists an element -v ? V, called the additive inverse of v, such that v + (-v) = 0. Distributivity of scalar multiplication with respect to vector addition s(v1 + v2) = sv1 + sv2. Distributivity of scalar multiplication with respect to field addition (n1 + n2)v = n1v + n2v. Respect of scalar multiplication over field's multiplication n1 (n2 s) = (n1 n2)s [nb 2] Identity element of scalar multiplication 1s = s, where 1 denotes the multiplicative identity in F.