GATE 2021 MA – Question 29

Linear Algebra · Vector Spaces and Linear Transformations · 1 mark · Numerical answer

T:ℝ³→ℝ³ is linear and T(1,1,1)=(1,−1,1),T²(1,1,1)=(1,1,1),T²(1,1,2)=(1,1,1). Find rank T.

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Correct answer: 2 to 2

Explanation

Write v=(1,1,1),w=(1,−1,1),a=(1,1,2). Then Tv=w and Tw=v, so the independent vectors v,w belong to range T and rank≥2. Also T²(a−v)=0 with a−v≠0, so T cannot be invertible. Hence rank≤2 and **rank T=2**.

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