MAT3701 ASSIGNMENT 1 2023

MAT3701
ASSIGNMENT 1
2023

,Solution:



1.1).



𝑊1 = {𝑋 ∈ 𝑀2×2 (𝐶): 𝐴𝑋 = 𝑋𝐵}


0 1 𝑖 0
𝐴=[ ] 𝑎𝑛𝑑 𝐵 = [ ]
−1 0 0 −𝑖
𝑎 𝑏
𝐿𝑒𝑡 ∶ 𝑋 = [ ] , 𝑡ℎ𝑒𝑛
𝑐 𝑑


𝐴𝑋 = 𝑋𝐵
0 1 𝑎 𝑏 𝑎 𝑏 𝑖 0
[ ][ ]=[ ][ ]
−1 0 𝑐 𝑑 𝑐 𝑑 0 −𝑖
𝑐 𝑑 𝑎𝑖 −𝑏𝑖
[ ]=[ ]
−𝑎 −𝑏 𝑐𝑖 −𝑑𝑖


𝑊𝑒 ℎ𝑎𝑣𝑒 ∶ 𝑐 = 𝑎𝑖
𝑑 = −𝑏𝑖

−𝑎 = 𝑐𝑖 ⇒ −𝑎𝑖 = 𝑐𝑖 2 ⇒ −𝑎𝑖 = −𝑐 ⇒ 𝑐 = 𝑎𝑖

−𝑏 = −𝑑𝑖 ⇒ −𝑏𝑖 = −𝑑𝑖 2 ⇒ −𝑏𝑖 = 𝑑 ⇒ 𝑑 = −𝑏𝑖


𝑎 𝑏
𝑋=[ ]
𝑐 𝑑
𝑎 𝑏
𝑋=[ ]
𝑎𝑖 −𝑏𝑖
1 0 0 1
𝑋 = 𝑎[ ]+𝑏[ ]
𝑖 0 0 −𝑖

, 𝐻𝑒𝑛𝑐𝑒
1 0 0 1
𝑇ℎ𝑒 𝑏𝑎𝑠𝑖𝑠 𝑜𝑓 𝑊1 = {[ ],[ ]}
𝑖 0 0 −𝑖


1.2).



𝑊2 = {𝑋 ∈ 𝑀2×2 (𝐶): 𝐴𝑋 = 𝑖𝑋}


0 1 𝑖 0
𝐴=[ ] 𝑎𝑛𝑑 𝐵 = [ ]
−1 0 0 −𝑖
𝑎 𝑏
𝐿𝑒𝑡 ∶ 𝑋 = [ ] , 𝑡ℎ𝑒𝑛
𝑐 𝑑


𝐴𝑋 = 𝑖𝑋
0 1 𝑎 𝑏 𝑎 𝑏
[ ][ ] = 𝑖[ ]
−1 0 𝑐 𝑑 𝑐 𝑑
𝑐 𝑑 𝑎𝑖 𝑏𝑖
[ ]=[ ]
−𝑎 −𝑏 𝑐𝑖 𝑑𝑖


𝑊𝑒 ℎ𝑎𝑣𝑒 ∶ 𝑐 = 𝑎𝑖
𝑑 = 𝑏𝑖
−𝑎 = 𝑐𝑖 ⇒ −𝑎𝑖 = 𝑐𝑖 2 ⇒ −𝑎𝑖 = −𝑐 ⇒ 𝑐 = 𝑎𝑖

−𝑏 = 𝑑𝑖 ⇒ −𝑏𝑖 = 𝑑𝑖 2 ⇒ −𝑏𝑖 = −𝑑 ⇒ 𝑑 = 𝑏𝑖


𝑎 𝑏
𝑋=[ ]
𝑐 𝑑
𝑎 𝑏
𝑋=[ ]
𝑎𝑖 𝑏𝑖
1 0 0 1
𝑋 = 𝑎[ ]+𝑏[ ]
𝑖 0 0 𝑖


𝐹𝑜𝑟 ∶ 𝑋 ∈ 𝑊1 ∩ 𝑊2 𝑖𝑡 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑡ℎ𝑎𝑡 ,
𝑋 ∈ 𝑊1 𝑎𝑛𝑑 ∶ 𝑋 ∈ 𝑊2


𝑎 𝑏 𝑎 𝑏
𝑋=[ ] 𝑎𝑛𝑑 𝑋 = [ ]
𝑎𝑖 −𝑏𝑖 𝑎𝑖 𝑏𝑖