MAT2611 Assignment TWO 2025

MAT2611 ASSIGNMENT TWO 2025

Problem 1




(1)

𝑈 = {(𝑥, 𝑦, 1) ∈ ℝ3 } under standard addition is not a vector space.

Let 𝒖 = (𝑥1 , 𝑦1 , 1) , 𝒗 = (𝑥2 , 𝑦2 , 1) , 𝒘 = (𝑥3 , 𝑦3 , 1), 𝑎, 𝑏 ∈ ℝ

VS1 is not satisfied:

𝒖 + 𝒗 = (𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 2) ∉ 𝑈

VS4 is not satisfied:

𝟎 is supposed to be (0, 0, 0) but (0, 0, 0) ∉ 𝑈

VS5 is not satisfied:

𝒖 = (𝑥1 , 𝑦1 , 1) ∈ 𝑈 but
−𝒖 = (−𝑥1 , −𝑦1 , −1) ∉ 𝑈

, VS6 is not satisfied:

𝒖 = (𝑥1 , 𝑦1 , 1) ∈ 𝑈 𝑎𝑛𝑑 3 ∈ ℝ but

3𝒖 = (3𝑥1 , 3𝑦1 , 3) ∉ 𝑈

(2)

𝑉 = {(𝑥, 𝑦, 0) ∈ ℝ3 } under standard addition is a vector space.

Let 𝒖 = (𝑥1 , 𝑦1 , 0) , 𝒗 = (𝑥2 , 𝑦2 , 0) , 𝒘 = (𝑥3 , 𝑦3 , 0), 𝑎, 𝑏 ∈ ℝ



VS1 is satisfied:

𝒖 + 𝒗 = (𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0 + 0) = ( 𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0) ∈ 𝑉

VS2 is satisfied:

𝒖 + 𝒗 = (𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0 + 0) = ( 𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0)

𝒗 + 𝒖 = (𝑥2 + 𝑥1 , 𝑦2 + 𝑦1 , 0 + 0) = ( 𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0) (the set of real numbers is
commutative under addition)
So, 𝒖 + 𝒗 = 𝒗 + 𝒖, ∀ 𝒖, 𝒗 ∈ 𝑉

VS3 is satisfied:

𝒖 + (𝒗 + 𝒘) = (𝑥1 , 𝑦1 , 0) + (𝑥2 + 𝑥3 , 𝑦2 + 𝑦3 , 0 + 0)

𝒖 + (𝒗 + 𝒘) = (𝑥1 , 𝑦1 , 0) + (𝑥2 + 𝑥3 , 𝑦2 + 𝑦3 , 0)

𝒖 + (𝒗 + 𝒘) = (𝑥1 + 𝑥2 + 𝑥3 , 𝑦1 + 𝑦2 + 𝑦3 , 0 + 0)

𝒖 + (𝒗 + 𝒘) = (𝑥1 + 𝑥2 + 𝑥3 , 𝑦1 + 𝑦2 + 𝑦3 , 0)


(𝒖 + 𝒗) + 𝒘 = (𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0 + 0) + ( 𝑥3 ,𝑦3 , 0)

(𝒖 + 𝒗) + 𝒘 = (𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 , 0) + ( 𝑥3 , 𝑦3 , 0)

(𝒖 + 𝒗) + 𝒘 = (𝑥1 + 𝑥2 + 𝑥3 , 𝑦1 + 𝑦2 + 𝑦3 , 0 + 0)

(𝒖 + 𝒗) + 𝒘 = (𝑥1 + 𝑥2 + 𝑥3 , 𝑦1 + 𝑦2 + 𝑦3 , 0)