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GATE 2020 MA – Question 51

Linear Algebra · Vector Spaces and Linear Transformations · 2 marks · Multiple choice

Let W₁,W₂,W₃ be subspaces of a finite-dimensional real vector space V. Which statement is TRUE?

  1. If $W_1+W_2+W_3=V$, then $\operatorname{span}(W_1\cup W_2)\cup\operatorname{span}(W_2\cup W_3)\cup\operatorname{span}(W_3\cup W_1)=V$.
  2. If $W_1\cap W_2=\{0\}=W_1\cap W_3$, then $W_1\cap(W_2+W_3)=\{0\}$.
  3. If $W_1+W_2=W_1+W_3$, then $W_2=W_3$.
  4. If $W_1\ne V$, then $\operatorname{span}(V\setminus W_1)=V$.

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Correct answer: (D) If $W_1\ne V$, then $\operatorname{span}(V\setminus W_1)=V$.

Explanation

Take v outside proper subspace W₁. For every w∈W₁, both v and v + w lie outside W₁, so $w=(v+w)-v$ belongs to the span of its complement. The complement itself is already in that span, so all V is spanned, proving D. The other statements incorrectly replace a subspace sum by a union or assume cancellation of subspace sums.

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