a linear transformation is defined to be orthogonal if ∥t(v)∥=∥v∥use the aabove to prove : suppose t: r2->r2 then T is orthogonal if ∥t(e1) ∥=∥T(e2)∥=1 and t(e1) × t(e2)=0. e1(1,0) e2(0,1)
please steps by steps
只要證明 "if"? 那應該很簡單吧?
又: t(e1) × t(e2)=0 的 "×" 應該是 inner product (或 dot product) 吧?
以下我還是用慣用符號 "．" 表示 dot product.
令 v = a e1 + b e2, 則
t(v) = a t(e1) + b t(e2).
所以
∥t(v)∥^2 = t(v)．t(v) = (a t(e1) + b t(e2))．(a t(e1) + b t(e2))
= a^2 ∥t(e1)∥^2 + b^2 ∥t(e2)∥^2 + 2ab t(e1)．t(e2)
= a^2 + b^2 = ∥v∥^2.
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只要證明 "if"? 那應該很簡單吧?
又: t(e1) × t(e2)=0 的 "×" 應該是 inner product (或 dot product) 吧?
以下我還是用慣用符號 "．" 表示 dot product.
令 v = a e1 + b e2, 則
t(v) = a t(e1) + b t(e2).
所以
∥t(v)∥^2 = t(v)．t(v) = (a t(e1) + b t(e2))．(a t(e1) + b t(e2))
= a^2 ∥t(e1)∥^2 + b^2 ∥t(e2)∥^2 + 2ab t(e1)．t(e2)
= a^2 + b^2 = ∥v∥^2.