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Part CCSIR NET June 2025k-dimensional-and-complementary-dimensional-subspaces-are-equinumerous

K dimensional and complementary dimensional subspaces are equinumerous

Consider the field 𝔽3_{3} consisting of 3 elements. Let V be an 𝔽3−_{3}-vector space of dimension 3 and W an 𝔽3−_{3}-vector space of dimension 2. Which of the following statements are true?

  1. A.The number of two dimensional subspaces of V is 13.✓ Correct
  2. B.The number of surjective linear transformations from V to W is 624.✓ Correct
  3. C.The number of one dimensional subspaces of V is 13.✓ Correct
  4. D.The number of linear transformations from V to W is 363^{6}.✓ Correct

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 standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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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