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

The zero functional is in the space

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

A.If f ∈ B, then dim ker f = 3
B.dim B = 3✓
C.dim A = 4✓
D.If f ∈ A, then the image of f is a one-dimensional $Q -$ 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.

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50 are analysed free — try those first.

The zero functional is in the space

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