Part BCSIR NET December 2024the-dual-of-a-span-not-of-the-ambient-space

The dual of a span not of the ambient space

Let U denote the span of {eᵗ, $e^{2}$ ᵗ, $e^{3}$ ᵗ} in the real vector space of continuous functions from $R$ to $R$ . Consider the $R -$ vector spaces V = { $f : U \to R$ | f is an $R -$ linear transformation} and W = {f ∈ V | $f (e^{3}$ ᵗ) = 0}. Which of the following statements is true?

A.Both V and W are infinite-dimensional
B.dim V = 3 and dim W = 1
C.dim V = 3 and dim W = 2✓
D.V is infinite-dimensional and dim W = 0

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 dual of a span not of the ambient space

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