Change of inner product

Let $⟨ \cdot | \cdot ⟩$ and $(\cdot | \cdot)$ be inner products on a vector space $𝑉$ . Let ${𝑣 𝑗}$ and ${𝑤 𝑗}$ be orthonormal bases with respect to $⟨ \cdot | \cdot ⟩$ and $(\cdot | \cdot)$ , and $𝑆 : 𝑉 \to 𝑉$ be a change of basis such that $𝑆 𝑤 𝑗 = 𝑣 𝑗$ . Then $⟨ 𝑆 𝑣 | 𝑆 𝑤 ⟩ = (𝑣 | 𝑤)$ for all $𝑣, 𝑤 \in 𝑉$ . #m/thm/linalg

Proof

Let $𝑣 = \sum 𝑗 𝛼 𝑗 𝑤 𝑗$ and $𝑤 = \sum 𝑗 𝛽 𝑗 𝑤 𝑗$ . Then

⟨ 𝑆 𝑣 | 𝑆 𝑤 ⟩ = ⟨ 𝑆 \sum 𝑗 𝛼 𝑗 𝑤 𝑗 | 𝑆 \sum 𝑘 𝛽 𝑘 𝑤 𝑘 ⟩ = ⟨ \sum 𝑗 𝛼 𝑗 𝑣 𝑗 | \sum 𝑘 𝛼 𝑘 𝑣 𝑘 ⟩ = \sum 𝑗, 𝑘 ――― 𝛼 𝑗 𝛽 𝑘 ⟨ 𝑣 𝑗 | 𝑣 𝑘 ⟩ = \sum 𝑗, 𝑘 ――― 𝛼 𝑗 𝛽 𝑘 𝛿 𝑗 𝑘 = \sum 𝑗, 𝑘 ――― 𝛼 𝑗 𝛽 𝑘 (𝑤 𝑗 | 𝑤 𝑘) = (\sum 𝑗 𝛼 𝑗 𝑤 𝑗 | \sum 𝑘 𝛼 𝑘 𝑤 𝑘) = (𝑣 | 𝑤)

as required.

#state/tidy| #lang/en | #SemBr