MAT2611 Assignment 2 Solutions 2026
UNISA
Due date: Friday, 29 May 2026
, Problem 1
Determine whether 𝑈and 𝑉are subspaces of ℝ4 .
(a) Subspace 𝑼
𝑈 = {(𝑥, 𝑦, 𝑧, 𝑢) ∈ ℝ4 : 𝑥 + 𝑦 − 𝑧𝑢 = 0}
To be a subspace, the set must satisfy:
1. Contains the zero vector
2. Closed under addition
3. Closed under scalar multiplication
Check the zero vector
Take (0, 0, 0, 0).
Substitute into the condition:
0 + 0 − (0)(0) = 0
True.
So the zero vector belongs to 𝑈.
Check closure under addition
Let
𝑎 = (𝑥1 , 𝑦1 , 𝑧1 , 𝑢1 ) ∈ 𝑈
and
𝑏 = (𝑥2 , 𝑦2 , 𝑧2 , 𝑢2 ) ∈ 𝑈
Then
𝑥1 + 𝑦1 − 𝑧1 𝑢1 = 0
and
𝑥2 + 𝑦2 − 𝑧2 𝑢2 = 0
UNISA
Due date: Friday, 29 May 2026
, Problem 1
Determine whether 𝑈and 𝑉are subspaces of ℝ4 .
(a) Subspace 𝑼
𝑈 = {(𝑥, 𝑦, 𝑧, 𝑢) ∈ ℝ4 : 𝑥 + 𝑦 − 𝑧𝑢 = 0}
To be a subspace, the set must satisfy:
1. Contains the zero vector
2. Closed under addition
3. Closed under scalar multiplication
Check the zero vector
Take (0, 0, 0, 0).
Substitute into the condition:
0 + 0 − (0)(0) = 0
True.
So the zero vector belongs to 𝑈.
Check closure under addition
Let
𝑎 = (𝑥1 , 𝑦1 , 𝑧1 , 𝑢1 ) ∈ 𝑈
and
𝑏 = (𝑥2 , 𝑦2 , 𝑧2 , 𝑢2 ) ∈ 𝑈
Then
𝑥1 + 𝑦1 − 𝑧1 𝑢1 = 0
and
𝑥2 + 𝑦2 − 𝑧2 𝑢2 = 0
