Mathematics · Linear Algebra

Let T: ℝ 4 → ℝ 4 be a linear map with four distinct eigenvalues and satisfying T 4 - 15T 2 + 10T + 24I = 0. Which of the following statements is necessarily true?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1T is a one-to-one function✓
2For every non-zero vector v &in; ℝ 4 , the set {2v, 3Tv} is linearly independent
3There exists a non-zero vector v 2 &in; ℝ 4 such that Tv 2 = v 2
4There exists a non-zero vector v 1 &in; ℝ 4 such that Tv 1 = 2v 1

Correct answer

T is a one-to-one function

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Let T: ℝ 4 &rarr; ℝ 4 be a linear map with four distinct eigenvalues and satisfying T 4 - 15T 2 + 10T + 24I = 0. Which of the following statements is necessarily true?