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Part BCSIR NET June 2025a-three-dimensional-image-leaves-the-kernel-infinite-dimensional

A three dimensional image leaves the kernel infinite dimensional

Let X be the R−\mathbb{R}-vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map φ:X→R3\varphi : X \to \mathbb{R}^{3} defined by φ(f)=(f(1),f′(1),f′′(1))\varphi(f) = (f(1), f'(1), f''(1)). Which of the following statements is true?

  1. A.The dimension of X/ker⁡φX/\ker \varphi is 3.✓ Correct
  2. B.ker⁡φ\ker \varphi is finite dimensional.
  3. C.The dimension of X/ker⁡φX/\ker \varphi is 1.
  4. D.X is finite dimensional.

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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