vector spaces

Question 1

closure under addition

Answer 1

u+v is a vector in V, axiom 1

Question 2

commutative property of addition

Answer 2

u+v=v+u, axiom 2

Question 3

associative property of addition

Answer 3

(u+v)+w=u+(v+w), axiom 3

Question 4

additive identity (zero element)

Answer 4

u+0=0+u=u, axiom 4

Question 5

additive inverse

Answer 5

u+(-u)=0, axiom 5

Question 6

closure under scalar multiplication

Answer 6

cu is a vector in V, axiom 6

Question 7

distributive property (scalars)

Answer 7

c(u+v)=cu+cv, axiom 7

Question 8

distributive property (vectors)

Answer 8

(c+d)u=cu+du (where c and d are scalars and u is a vector), axiom 8

Question 9

associative properties of scalar multiplication

Answer 9

c(du)=(cd)u (where c and d are scalars and u is a vector), axiom 9

Question 10

scalar identity

Answer 10

1u=u, axiom 10

Question 11

axioms involving sets

Answer 11

axioms 1,4 and 6, closure under addition, additive identity (zero element), closure under scalar multiplication

Question 12

axioms involving vector addition

Answer 12

all axioms 1-5, closure under addition, commutative property of addition, associative property of addition, additive identity (zero element), additive inverse

Question 13

axioms involving scalar multiplication

Answer 13

all axioms 6-10, closure under scalar multiplication, distributive property (scalars), distributive property (vectors), associative properties of scalar multiplication, scalar identity