MAT2611 Sem ONE Assign THREE 2022

MAT2611 SEMESTER 1 ASSIGNMENT 3 2022

Problem 7




𝑆 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 ) ∈ ℝ5 ∶ 𝑥1 = 3𝑥2 𝑎𝑛𝑑 𝑥3 = 7𝑥4 }

1.
0ℝ5 = (0, 0, 0, 0, 0)

𝑂𝑛 0ℝ5 , 𝑧1 = 0, 𝑧2 = 0, 𝑧3 = 0, 𝑧4 = 0 𝑎𝑛𝑑 𝑧5 = 0

𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡:
3𝑧2 = 3(0) = 0 = 𝑧1
7𝑧4 = 7(0) = 0 = 𝑧3

𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑡ℎ𝑒 𝑧𝑒𝑟𝑜 𝑜𝑓 ℝ5 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑆.

𝑆 𝑖𝑠 𝑎 𝑛𝑜𝑛𝑒𝑚𝑝𝑡𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 ℝ5

2.
𝐿𝑒𝑡 𝑥, 𝑦 ∈ 𝑆
𝑥 = (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 ) 𝑤𝑖𝑡ℎ 𝑥1 = 3𝑥2 𝑎𝑛𝑑 𝑥3 = 7𝑥4

𝑦 = (𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 , 𝑦5 ) 𝑤𝑖𝑡ℎ 𝑦1 = 3𝑦2 𝑎𝑛𝑑 𝑦3 = 7𝑦4

𝑥 + 𝑦 = (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 ) + (𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 , 𝑦5 )

𝑥 + 𝑦 = (𝑥1 + 𝑦1 , 𝑥2 + 𝑦2 , 𝑥3 + 𝑦3 , 𝑥4 + 𝑦4 , 𝑥5 + 𝑦5 )

𝑥1 + 𝑦1 = 3𝑥2 + 3𝑦2
𝑥1 + 𝑦1 = 3(𝑥2 + 𝑦2 )

𝑥3 + 𝑦3 = 7𝑥4 + 7𝑦4
𝑥3 + 𝑦3 = 7(𝑥4 + 7𝑦4 )

𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑥 + 𝑦 ∈ 𝑆

𝑆 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛