Linear Algebra Done Right · Problem 7A.30

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Suppose that T∈L(F3)T \in \mathcal{L}(\mathbf{F}^3) is normal and T(1,1,1)=(2,2,2)T(1, 1, 1) = (2, 2, 2). Suppose (z1,z2,z3)∈null⁡T(z_1, z_2, z_3) \in \operatorname{null} T. Prove that z1+z2+z3=0z_1 + z_2 + z_3 = 0.

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