CHE2611
ASSIGNMENT 2
2025
, QUESTION 1
1).
U = {(x, y, 1) ∈ R3 }
∴ (0,0,0) ∉ U
Since U does not contain the zero vector, then U is not a vector space.
2).
V = {(x, y, 1) ∈ R3 }
If x = 0 and y = 0, we have
(0,0,0) ∈ V (Contains the zero vector)
Suppose u1 , u2 ∈ V
Let u1 = (x1 , y1 , 0) and u2 = (x2 , y2 , 0)
u1 + u2 = (x1 , y1 , 0) + (x2 , y2 , 0)
= (x1 + x2 , y1 + y2 , 0 + 0)
= (x1 + x2 , y1 + y2 , 0) ∈ V (Closed under addition)
Suppose we have a scalar λ ∈ F:
λ ∙ u1 = λ(x1 , y1 , 0)
= (λx1 , λy1 , 0) ∈ V (Closed under addition)
V is a vector space.
ASSIGNMENT 2
2025
, QUESTION 1
1).
U = {(x, y, 1) ∈ R3 }
∴ (0,0,0) ∉ U
Since U does not contain the zero vector, then U is not a vector space.
2).
V = {(x, y, 1) ∈ R3 }
If x = 0 and y = 0, we have
(0,0,0) ∈ V (Contains the zero vector)
Suppose u1 , u2 ∈ V
Let u1 = (x1 , y1 , 0) and u2 = (x2 , y2 , 0)
u1 + u2 = (x1 , y1 , 0) + (x2 , y2 , 0)
= (x1 + x2 , y1 + y2 , 0 + 0)
= (x1 + x2 , y1 + y2 , 0) ∈ V (Closed under addition)
Suppose we have a scalar λ ∈ F:
λ ∙ u1 = λ(x1 , y1 , 0)
= (λx1 , λy1 , 0) ∈ V (Closed under addition)
V is a vector space.
