i am a university student starting on a linear algebra course and i am struggling with some vector spaces questions. Help would be greatly appreciated and solutions written with full steps would be very much helpful. Thanks.
1. Let V denote the set of ordered pairs of real numbers. If (a1, a2) and (b1, b2) are elements of V and c ∈ R (all real numbers), define
(a1, a2) + (b1, b2) = (a1 + b1, a2b2), c(a1, a2)=(ca1, a2)
Is V a vector space with these operations? (check all 8 axioms)
2. Let V and W be vector spaces. Let Z = {(v,w)| v∈V, w∈W}. Prove Z is a vector space with the operations
(v1, w1) + (v2, w2) = (v1 + v2, w1 + w2), c(v1,w1)= (cv1, cw1).
Check axioms 1-4. Space Z is known as product V x W.
*my problem with these questions arise from the fact that it is simple to prove VS1 but I have no idea how to prove the other axioms like VS 2 or VS 4.
1. Let V denote the set of ordered pairs of real numbers. If (a1, a2) and (b1, b2) are elements of V and c ∈ R (all real numbers), define
(a1, a2) + (b1, b2) = (a1 + b1, a2b2), c(a1, a2)=(ca1, a2)
Is V a vector space with these operations? (check all 8 axioms)
2. Let V and W be vector spaces. Let Z = {(v,w)| v∈V, w∈W}. Prove Z is a vector space with the operations
(v1, w1) + (v2, w2) = (v1 + v2, w1 + w2), c(v1,w1)= (cv1, cw1).
Check axioms 1-4. Space Z is known as product V x W.
*my problem with these questions arise from the fact that it is simple to prove VS1 but I have no idea how to prove the other axioms like VS 2 or VS 4.
