Kunze Linear Algebra · Problem 3.5.1

Linear Transformations

Mathematics — Methods · linear-functionals-computation

Problem

In R3R^3, let α1=(1,0,1)\alpha_1 = (1, 0, 1), α2=(0,1,−2)\alpha_2 = (0, 1, -2), α3=(−1,−1,0)\alpha_3 = (-1, -1, 0). (a) If ff is a linear functional on R3R^3 such that f(α1)=1,f(α2)=−1,f(α3)=3,f(\alpha_1) = 1, \quad f(\alpha_2) = -1, \quad f(\alpha_3) = 3, and if α=(a,b,c)\alpha = (a, b, c), find f(α)f(\alpha). (b) Describe explicitly a linear functional ff on R3R^3 such that f(α1)=f(α2)=0butf(α3)≠0.f(\alpha_1) = f(\alpha_2) = 0 \quad \text{but} \quad f(\alpha_3) \neq 0. (c) Let ff be any linear functional such that f(α1)=f(α2)=0andf(α3)≠0.f(\alpha_1) = f(\alpha_2) = 0 \quad \text{and} \quad f(\alpha_3) \neq 0. If α=(2,3,−1)\alpha = (2, 3, -1), show that f(α)≠0f(\alpha) \neq 0.

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