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GATE 2020 EC – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

If $\mathbf v_1,\mathbf v_2,\dots,\mathbf v_6$ are six vectors in $\mathbb R^4$, which one of the following statements is FALSE?

  1. It is not necessary that these vectors span $\mathbb R^4$.
  2. These vectors are not linearly independent.
  3. Any four of these vectors form a basis for $\mathbb R^4$.
  4. If $\{\mathbf v_1,\mathbf v_3,\mathbf v_5,\mathbf v_6\}$ spans $\mathbb R^4$, then it forms a basis for $\mathbb R^4$.

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Correct answer: (C) Any four of these vectors form a basis for $\mathbb R^4$.

Explanation

Six vectors in a 4-dimensional space are always linearly dependent (B is true), and they need not span the space (A is true). Four vectors that span $\mathbb R^4$ form a basis (D is true). But an arbitrary set of four of the six may be linearly dependent, so C is false.