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Suppose $V$ and $W$ are finite-dimensional vector spaces, and that $\phi : V \to W$ is a linear map that is injective but not surjective. What can you say about the dimensions of $V$ and $W$?

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closed as off-topic by Rory Daulton, user296602, Michael Albanese, Shahab, choco_addicted Mar 17 '16 at 2:35

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By the Rank-Nullity Theorem, $$ \dim V = \dim \ker \varphi + \dim \operatorname{im} \varphi = \dim \operatorname{im} \varphi < \dim W $$

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Let $\{e_1,\ldots,e_n\}$ be a basis of $V$. Then $\{\phi(e_1),\ldots,\phi(e_n)\}$ is linear independent in $W$ (easy to show). Thus, $\dim W\ge n = \dim V$.

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