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Part CCSIR NET December 2023union-of-subspaces

Union of subspaces

Let 𝔽 be a finite field and V be a finite dimensional non-zero 𝔽-vector space. Which of the following can NEVER be true?

  1. A.V is the union of 2 proper subspaces.
  2. B.V is the union of 3 proper subspaces.
  3. C.V has a unique basis.
  4. D.V has precisely two bases.

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The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: An injective linear operator on a vector space is surjective

More on this topic

From Vector Spaces and Linear Maps › Bases, dimension, rank–nullity

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