Let f map C2 into C2 be a linear operator such that for every x in C2, f(f(x)) = x. Let V+ = {x in C2 | f(x) = x} and V- = {x in C2 | f(x) = -x}. Prove that every x in C2 can be written as x = x+ + x- , where x+ is in V+ and x- is in V- .

Let f map C2 into C2 be a linear operator such that for every x in C2, f(f(x)) = x. Let V+ = {x in C2 | f(x) = x} and V- = {x in C2 | f(x) = -x}. Prove that every x in C2 can be written as x = x+ + x- , where x+ is in V+ and x- is in V- .

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 4E: 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .

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Let f map C² into C² be a linear operator such that for every x in C², f(f(x)) = x. Let V₊ = {x in C² | f(x) = x} and V_- = {x in C² | f(x) = -x}. Prove that every x in C² can be written as x = x₊ + x_- , where x₊ is in V₊ and x_- is in V_- .

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