Suppose that we have two finite geberated vector spaces $V$ and $S$ over a field $\mathbb{K}$. Let $\phi:V \rightarrow S$ a function. What does it mean that $\phi$ is an embedding?
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$\phi$ is a vector-space homomorphism which is injective. That is, $\phi$ gives us a way of viewing elements of $V$ as elements of $S$.
Patrick Stevens
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Let me give you an example. In my example i have an application $\phi:V \rightarrow S$ injective. If i prove that the dimension of $S$ is the same of the dimension of $V$ can i say that $\phi$ is a bijection? – dario Aug 22 '15 at 12:31
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Yes, if it's an embedding. Vector spaces of the same dimension with the same underlying field are isomorphic. (https://math.stackexchange.com/questions/1025926/vector-spaces-of-the-same-finite-dimension-are-isomorphic) Then use https://math.stackexchange.com/questions/200774/question-on-finite-vector-spaces-injective-surjective-and-if-v-is-not-finite . – Patrick Stevens Aug 22 '15 at 12:35