Why is the group $GL(n, \mathbb{R})$ of dimension $n^{2}$?
Dimension of $GL(n, \mathbb{R})$
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$\begingroup$
linear-algebra
matrices
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1The group $M(n, \mathbb{R})$ of $n\times n$ matrices has dimension $n^2$. Why? It's easy to show that $M(n, \mathbb{R})$ a vector space. Any element $\in M(n, \mathbb{R})$ can be written in terms of a sum of $n^2$ matrices $B_{ij}$ where the entry $i,j)$ of $B_{ij} = 1$, and zero otherwise. You can show that the set $\{B_{ij}\}$ is a basis of $M(n, \mathbb{R})$, and hence $M(n, \mathbb{R})$ has dimension $n^2$. $GL(n,\mathbb{R})$ is a subset of $M(n, \mathbb{R})$ under the determinant map. It has the same dimension by Steve's answer below. – 2012-03-06
2 Answers
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It is an open submanifold of the set of $n$ by $n$ matrices, which is a vector space of dimension $n^2$.
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0Is there are more elementary way to see this though? – 2012-03-06
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7@198203: Isn't that plenty elementary already? It's about the most elementary imaginable argument that can reach the relevant definition of "dimension" at all. – 2012-03-06
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1How many basis elements does an $n$ by $n$ matrix have? Certainly it must be $n^2$... take the obvious basis: a single entry in each matrix to be $1$ and all others to be $0$. Do this $n \cdot n$ times, one for each position in the matrix. – 2012-03-06
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0Maybe I'm just not seeing it, but why does an open submanifold of a vector space must have the dimension of that vector space? – 2012-03-06
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0Think of an open ball in $\mathbb R^2$, perhaps that would help? This ball clearly has dimension $2$. Now the same ideas go for an open submanifold of $M(n,\mathbb R)$. – 2012-03-06
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2Basically, the determinant is a continuous function from the $n\times n$ matrices to $\mathbb R$, and the invertible ones are the ones with non-zero determinant, so $GL(n)=det^{-1}(\mathbb {R}\setminus 0)$. Since $\mathbb R\setminus 0$ is an open subset of $\mathbb R$, $GL(n)$ has to be an open subset of the set of all matrices. – 2012-03-06
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2@198203 Do you know the definition of the dimension of a manifold? Once you have the definition, the fact that an open submanifold of an $n$-manifold has dimension $n$ is essentially immediate. What's non-trivial and requires work is the proof that the notion of the dimension of a manifold is well-defined. – 2012-03-06
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0Woah, did not expect so many responses. Thanks math.SE!, it makes sense now. – 2012-03-07
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You should notice that the determinant is a continuous map $f:M(n,\mathbb{R})\equiv\mathbb{R}^{n^2}\to \mathbb{R}$, $f(X)=\det(X)$
Then note that
$$GL(n, \mathbb{R})=f^{-1}(\mathbb{R}\setminus\{0\})$$
How the pre-image by continuous map of open set is an open set then $GL(n, \mathbb{R})$ is an open set of $\mathbb{R}^{n^2}$ whose dimension is $n^2$.
(I am here using that an open set of $\mathbb{R}^{k}$ has $k$-dimension.)
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0How is, by the way, the dimension of an open set defined ? I am wondering because i am familiar with dimensions of vectorspaces etc. but not this. – 2012-11-19
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1If a manifold as a topological space is locally homeomorphic to $R^n$ then its dimension is $n$. We can also think about the Haussdorf Dimension. – 2012-11-23