Let V be a finite-dimensional complex inner product space. For a linear map T: V → V, let T denote its adjoint. Which of the following statements is false?

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Mathematics · Linear Algebra

Verified Solution4 OptionsUpdated: October 2026👁️1 View

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If T = -T (skew-adjoint) and N > 1 is an integer, and v ∈ V is such that TN(v) = 0, it is never true that T(v) = 0 unless v is the zero vector.

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Answer 1

If TT = TT (normal) and N > 1 is an integer, and v ∈ V is such that TN(v) = 0, then T(v) = 0.

Answer 2

If T = -T (skew-adjoint) and N > 1 is an integer, and v ∈ V is such that TN(v) = 0, it is never true that T(v) = 0 unless v is the zero vector.

Answer 3

If v ∈ V is such that TT(v) = 0, then T(v) = 0.

Answer 4

If the trace of TT is zero, then T = 0.

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