JC TUTORIALS
MAT2611
ASSIGNMENT 7
SEMESTER 1
2022
, Problem 23:
Solution:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡ℎ𝑎𝑡 𝜆 ∈ 𝐹 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑇
𝑆𝑖𝑛𝑐𝑒 𝜆 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑆𝑇 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑎 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 ∈ 𝑉 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
(𝑆𝑇)𝑣 = 𝜆𝑣
𝑁𝑜𝑤
(𝑇𝑆)(𝑇𝑣) = 𝑇(𝑆𝑇𝑣)
= 𝑇(𝜆𝑣)
= 𝜆𝑇𝑣
𝜆 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑇𝑆. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑆𝑇 𝑎𝑛𝑑 𝑇𝑆 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠.
Problem 24:
Solution:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝜆 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑇. 𝑇ℎ𝑒𝑛
MAT2611
ASSIGNMENT 7
SEMESTER 1
2022
, Problem 23:
Solution:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡ℎ𝑎𝑡 𝜆 ∈ 𝐹 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑇
𝑆𝑖𝑛𝑐𝑒 𝜆 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑆𝑇 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑎 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 ∈ 𝑉 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
(𝑆𝑇)𝑣 = 𝜆𝑣
𝑁𝑜𝑤
(𝑇𝑆)(𝑇𝑣) = 𝑇(𝑆𝑇𝑣)
= 𝑇(𝜆𝑣)
= 𝜆𝑇𝑣
𝜆 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑇𝑆. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑆𝑇 𝑎𝑛𝑑 𝑇𝑆 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠.
Problem 24:
Solution:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝜆 𝑖𝑠 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑇. 𝑇ℎ𝑒𝑛
