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

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

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

The chapter behind this: Bases, dimension and rank — free to read

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

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