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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, e2e^{2}ᵗ, e3e^{3}ᵗ} in the real vector space of continuous functions from R\mathbb{R} to R\mathbb{R}. Consider the R−\mathbb{R}-vector spaces V = {f:U→Rf : U \to \mathbb{R} | f is an R−\mathbb{R}-linear transformation} and W = {f ∈ V | f(e3f(e^{3}ᵗ) = 0}. Which of the following statements is true?

  1. A.Both V and W are infinite-dimensional
  2. B.dim V = 3 and dim W = 1
  3. C.dim V = 3 and dim W = 2✓ Correct
  4. 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, 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

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