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

K dimensional and complementary dimensional subspaces are equinumerous

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

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

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.

See pricing

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

K dimensional and complementary dimensional subspaces are equinumerous

More on this topic