The GATE Grind

GATE 2023 ST – Question 44

Matrix Theory · Vector Spaces and Linear Systems · 2 marks · Multiple choice

For U⊂ℝⁿ let L(U)=span(U). For T,S⊂ℝⁿ,which statement is NOT true?

  1. T properly contained in S implies L(T) properly contained in L(S).
  2. L(L(S))=L(S).
  3. L(T∪S)={u+v:u∈L(T),v∈L(S)}.
  4. If α+2β+3γ=0,then L({α,β})=L({β,γ}).

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Show answer and explanation

Correct answer: (A) T properly contained in S implies L(T) properly contained in L(S).

Explanation

Proper set inclusion need not enlarge a span: T={e₁},S={e₁,2e₁} have identical spans. B follows because a span is already a subspace,and C is the sum-of-subspaces identity. In D,α=−2β−3γ and γ=−(α+2β)/3 give both inclusions. Thus **A**.

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