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MAT2611 Sem ONE Assign SEVEN 2022

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MAT2611 Semester 1 ONE Assignment SEVEN of 2022

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MAT2611 SEMESTER 1 ASSIGNMENT 7 2022

Problem 23




𝑉 𝑖𝑠 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑝𝑎𝑐𝑒 𝑤𝑖𝑡ℎ 𝑑𝑖𝑚𝑉 = 𝑛

𝑆 𝑎𝑛𝑑 𝑇 𝑎𝑟𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟𝑠.

𝑆: 𝑉 → 𝑉 𝑎𝑛𝑑 𝑇: 𝑉 → 𝑉

𝐿𝑒𝑡 𝜆 𝑏𝑒 𝑎𝑛 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑇

𝑅𝑒𝑐𝑎𝑙𝑙 𝑡ℎ𝑎𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝑚𝑎𝑦 𝑏𝑒 𝑧𝑒𝑟𝑜 𝑏𝑢𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑚𝑎𝑦 𝑛𝑜𝑡 𝑏𝑒 𝑧𝑒𝑟𝑜 𝑣𝑒𝑐𝑡𝑜𝑟𝑠.

⇒ 𝑇ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 ∈ 𝑉 𝑤𝑖𝑡ℎ 𝑣 ≠ 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑆𝑇 (𝑣) = 𝜆𝑣

⇒ 𝑆(𝑇(𝑣)) = 𝜆𝑣

𝐿𝑒𝑡 𝑦 = 𝑇(𝑣)

𝑆𝑖𝑛𝑐𝑒 𝑇 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑎𝑛𝑑
𝑆(𝑇(𝑣)) = 𝜆𝑣
𝑇ℎ𝑒𝑛 𝑇 (𝑆(𝑇(𝑣))) = 𝑇(𝜆𝑣)
⇒ 𝑇𝑆(𝑇(𝑣)) = 𝜆𝑇(𝑣)
⇒ 𝑇𝑆 (𝑦) = 𝜆𝑦

𝐶𝑎𝑠𝑒 1 (𝜆 ≠ 0)

𝜆≠0

⇒ 𝜆𝑣 ≠ 0

⇒ 𝑇(𝜆𝑣) ≠ 𝑇 (0)

⇒ 𝑇(𝜆𝑣) ≠ 0

⇒ 𝜆𝑇(𝑣) ≠ 0

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