Exercise 7.A.13

Answers

Proof. Define T ∈ L(ℂ4) by

T(z1,z2,z3,z4) = (z4,z1,z2,z3).

We have

⟨(z1,z2,z3,z4),T∗(x 1,x2,x3,x4)⟩ = ⟨T(z1,z2,z3,z4),(x1,x2,x3,x4)⟩ = ⟨(z4,z1,z2,z3),(x1,x2,x3,x4)⟩ = z4x1¯ + z1x2¯ + z2x3¯ + z3x4¯ = ⟨(z1,z2,z3,z4),(x2,x3,x4,x1)⟩.

Thus T∗(z1,z2,z3,z4) = (z2,z3,z4,z1). Note that T is normal (T∗T and TT∗ equal the identity), however T≠T∗. □

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2017-10-06 00:00
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