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Is this true?

An injective linear operator on a vector space is surjective

No — it is false.

The counterexample

The right shift on

Injective but misses everything with a non-zero first coordinate. Rank–nullity needs finite dimension.

The kind of mistake this is

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this
linear algebra

Others that fail the same way

From Vector Spaces and Linear Maps › Bases, dimension, rank–nullity

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