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I know it is a very simple question but I thought it would be useful for more people, since it is kind of confusing because of the similarity of the notation. From wikipedia, I was given that $GL(V)$ represents the group of isomorphisms that are biyective, while I know by myself that when written $f\in\mathcal{L}(V)$, it means that $f: V\rightarrow V$. Is the difference between both that $GL(V)$ is a group while $\mathcal{L}(V)$ is a set? Added into the fact that $f\in \mathcal{L}(V)$ doesn't necessarily need to be biyective, isn't it?

Aley20
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  • $\mathcal{L}(V)={\rm Hom}(V,V)={\rm End}(V)$ is more than a set. It is also called the endomorphism ring of $V$. For a vector space $V$ of dimension $n$ over a field $K$, we have the group $GL(V)\cong GL_n(K)$ and the endomorphism ring $\mathcal{L}(V)$. See for example this post. – Dietrich Burde Apr 18 '23 at 13:29
  • @DietrichBurde okay then what is the difference between a ring and a group? Also I haven't understood your last sentence, note I am kind of an starter to linear algebra – Aley20 Apr 18 '23 at 13:36
  • If this is your question, what the differences are between groups and rings (or fields), then see the duplicate. Also, note that $V$ is a vector space, but $GL(V)$ is not a vector space (zero is missing) and $\mathcal{L}(V)$ is also a vector space. – Dietrich Burde Apr 18 '23 at 13:38
  • @DietrichBurde I still don't understand what is $GL(V)$, I don't know why would anyone close this, the question still remains as I don't know the difference between them, I know now the difference between a ring and a group, but what does that have to do with the question about $GL(V)$? I was just asking about it to be curious – Aley20 Apr 18 '23 at 16:40
  • @DietrichBurde I still don't understand what is $GL(V)$, I don't know why would anyone close this, the question still remains as I don't know the difference between them, I know now the difference between a ring and a group, but what does that have to do with the question about $GL(V)$? I was just asking about it to be curious – Aley20 Apr 18 '23 at 16:40
  • What else do you want to know? We have discussed it, right? $GL(V)$ is as you said, the group of invertible linear maps $f\colon V\rightarrow V$. It can be realised by invertible matrices, i.e., as $GL_n(K)$. It is a group, and not a vector space. The other space is $\mathcal{L}(V)$, which is a vector space (and a ring and an algebra). I thought your main problem is the difference between these algebraic structures (groups, rings, vector spaces etc.), as you said in the comment. – Dietrich Burde Apr 18 '23 at 16:50

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