16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.6

Inner Product Spaces

Mathematics — Linear Algebra · inner-products

Problem

Let ( ∣ )( \ | \ ) be the standard inner product on R2R^2, and let TT be the linear operator T(x1,x2)=(−x2,x1)T(x_1, x_2) = (-x_2, x_1). Now TT is 'rotation through 90°90°' and has the property that (α∣Tα)=0(\alpha|T\alpha) = 0 for all α\alpha in R2R^2. Find all inner products [ ∣ ][ \ | \ ] on R2R^2 such that [α∣Tα]=0[\alpha|T\alpha] = 0 for each α\alpha.

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