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Suppose $V:=GL(n,K)$ denotes the general linear group and $V_I$ denotes center of $V$. Then what exactly does $V/V_I$ represent? I would be appreciate if you give a similar good example for better understanding. (The origin of this question is General projective transformations ${\rm PGL}(n,K)$ and special projective transformations ${\rm PSL}(n,K)$)

C.F.G
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    The identity element of a vector space is the zero vector. It spans a trivial subspace. In other words $V_I = {0}$ and $V / V_I$ is generated by the equivalence relation on $V$ where $u\sim v \iff u-v\in V_I$ i.e $u=v$. So for all intents and purposes $V = V / V_I$. – AlvinL Nov 22 '17 at 08:11
  • So why $PGL(n,K)=GL/ z$ where $z$ is the center of $GL(n,K)$. isn't center of $GL(n,K)$ equal to $V_I$? – C.F.G Nov 22 '17 at 09:02
  • The center of a group is the subset of those elements that commute with all elements in the group. What makes you think a center must always be trivial? – AlvinL Nov 22 '17 at 09:14
  • $PGL(n,K) = GL/z$ is merely the definition of what a projective linear group is (according to wiki). I don't understand what your question(s) are. – AlvinL Nov 22 '17 at 09:17
  • According to this post https://math.stackexchange.com/questions/960342/center-of-general-linear-group $V_I=a I$ – C.F.G Nov 22 '17 at 09:36
  • $I$ is the identity transformation of $V$ i.e $I:V\to V, x\mapsto x$. The space in question, therefore, is $V_I = {aI : a\in K}$. This is a subspace of $\mbox{End}(V)$. The meaning of $V/V_I$ is no longer clear to me. – AlvinL Nov 22 '17 at 11:49
  • By this question I want to know the structure of ${\rm PGL}(n,K)$ and ${\rm PSL}(n,K)$. maybe my question is wrong? – C.F.G Nov 26 '17 at 07:30

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