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

Cardinality of a hamel basis

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

A.There exists an injective $Q -$ linear transformation $T : R \to Q [x]$
B.There exists an injective $Q -$ linear transformation $T : Q [x] \to R$ ✓
C.The $Q -$ vector spaces $Q [x]$ and $R$ are isomorphic
D.There do not exist non-zero $Q -$ linear transformations $T : R \to Q [x]$

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.

50 are analysed free — try those first.

Related counterexample: An injective linear operator on a vector space is surjective

More on this topic

rank is field independentJune 2023
union of subspacesDecember 2023
the dual of a span not of the ambient spaceDecember 2024
row space complement is the null spaceDecember 2024
the zero functional is in the spaceDecember 2024
rank nullity inequality directionJune 2024

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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