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Homework week 12 1. Linea Vector Space: Let V a non-empty set and let K a field. We say that V is a linear vector space over the field K if two operations called addition and multiplication for a scalar are defined in such a way that the following axioms are satisfied a) Closure under Addition: The addition operation assigns to each ordered pair u, v where u v V a unique element of V u v w called the sum of u and v. b) Associativity under Addition: u, v, w V u v w u v w c) Identity Element under Addition: 0 V such that 0 v v for all v V . d) Inverse Element under Addition: v V v V v v 0 e) Commutativity under addition: u, v V v u u v f) Closure under Multiplication for a scalar: The multiplication for a scalar operation assigns to each ordered pair a, v where a K and v V a unique element of V av w called the product of a by v. g) Distributive law: a K; u v V a u v au av h) Distributive law: a b K; v V a b v av bv a b V ; v V a bv ab v i) Associativity under Multiplication for a scalar: j) Multiplication by the K’s unity: If 1 is the unity of K then 1v v for all v V . The elements of V are called generically vectors and the elements of K are called scalars. ——————————————————————————————————– 1 1. Decide if the the set V x y x y x y with F xy k x y and the following operation rules: addition: x x y y 2kx 2ky by checking if the axioms b,d,g,h and i are consistent. a 1 a b with F 1 b scalar multiple. Does not form a vector space. a 0 a b 3. Show that the set V 0 b scalar multiple. Does form a vector space. and the common operations of addition and with F and the common operations of addition and 2. Show that the set V 2