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.
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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