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Part CCSIR NET December 2024the-zero-functional-is-in-the-space

The zero functional is in the space

For a variable x, consider the Q−\mathbb{Q}-vector space V = {ax3+^{3} + bx2+^{2} + cx + d | a,b,c,d∈Qa, b, c, d \in \mathbb{Q}}. Further, let A = {f:V→Qf : V \to \mathbb{Q} | f is aQ−a \mathbb{Q}-linear transformation} and B = {f ∈ A | f(1) = 0}. Which of the following statements are true?

  1. A.If f ∈ B, then dim ker f = 3
  2. B.dim B = 3✓ Correct
  3. C.dim A = 4✓ Correct
  4. D.If f ∈ A, then the image of f is a one-dimensional Q−\mathbb{Q}-vector space.

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

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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