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

A.The dimension of $X / ker φ$ is 3.✓
B. $ker φ$ is finite dimensional.
C.The dimension of $X / ker φ$ is 1.
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

A three dimensional image leaves the kernel infinite dimensional

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