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Part CCSIR NET June 2024cardinality-of-a-hamel-basis

Cardinality of a hamel basis

Consider R\mathbb{R} and Q[x]\mathbb{Q}[x] as vector spaces over Q\mathbb{Q}. Which of the following statements are true?

  1. A.There exists an injective Q−\mathbb{Q}-linear transformation T:R→Q[x]T : \mathbb{R} \to \mathbb{Q}[x]
  2. B.There exists an injective Q−\mathbb{Q}-linear transformation T:Q[x]→RT : \mathbb{Q}[x] \to \mathbb{R}✓ Correct
  3. C.The Q−\mathbb{Q}-vector spaces Q[x]\mathbb{Q}[x] and R\mathbb{R} are isomorphic
  4. D.There do not exist non-zero Q−\mathbb{Q}-linear transformations T:R→Q[x]T : \mathbb{R} \to \mathbb{Q}[x]

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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