Part CCSIR NET June 2023kernel-dimension-vs-block-sizes

Kernel dimension vs block sizes

Let V be a finite dimensional real vector space and $T_{1}, T_{2}$ be two nilpotent operators on V. Let $W_{1} =$ { $v \in V : T_{1}(v) = 0$ } and $W_{2} =$ { $v \in V : T_{2}(v) = 0$ }. Which of the following statements are FALSE?

A.If $T_{1}$ and $T_{2}$ are similar, then $W_{1}$ and $W_{2}$ are isomorphic vector spaces.
B.If $W_{1}$ and $W_{2}$ are isomorphic vector spaces, then $T_{1}$ and $T_{2}$ have the same minimal polynomial.✓ Correct
C.If $W_{1} = W_{2} = V$ , then $T_{1}$ and $T_{2}$ are similar.
D.If $W_{1}$ and $W_{2}$ are isomorphic, then $T_{1}$ and $T_{2}$ have the same characteristic polynomial.

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 invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

Related counterexample: Every real matrix has a Jordan form over ℝ

Kernel dimension vs block sizes

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