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For each of the examples below find an operator T* on a vector space space V such that for all x, y in V (T(x), y) = (x, T* (y)), where (,) denotes the inner product on V. (a) V = C², standard inner product, T((2₁, 22)) = (2₁ + iz₁ − iz₂, iz₁ - 3z2+2iz₂). (h) V-ff R(R) SL

For each of the examples below find an operator T* on a vector space space V such that for all x, y in V (T(x), y) = (x, T* (y)), where (,) denotes the inner product on V. (a) V = C², standard inner product, T((2₁, 22)) = (2₁ + iz₁ − iz₂, iz₁ - 3z2+2iz₂). (h) V-ff R(R) SL

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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16. For each of the examples below find an operator T* on a vector space space V such that for all x, y in V (T(x), y) = (x, T* (y)), where (,) denotes the inner product on V. (a) V = C², standard inner product, T((21, 22)) = (z₁ + iz₁ - izz, iz₁ - 3z2 + 2iz₂). (b) V = {ƒ € P(R) : ƒ(−1) = f(1) = 0}, (f,g) = f¹₁f(x) g(x) dx, T(ƒ) = f' + ƒ. Hint for (b): Integrate by parts.
Transcribed Image Text:16. For each of the examples below find an operator T* on a vector space space V such that for all x, y in V (T(x), y) = (x, T* (y)), where (,) denotes the inner product on V. (a) V = C², standard inner product, T((21, 22)) = (z₁ + iz₁ - izz, iz₁ - 3z2 + 2iz₂). (b) V = {ƒ € P(R) : ƒ(−1) = f(1) = 0}, (f,g) = f¹₁f(x) g(x) dx, T(ƒ) = f' + ƒ. Hint for (b): Integrate by parts.
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