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Why is the group $GL(n, \mathbb{R})$ of dimension $n^{2}$?

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The 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. –  user2468 Mar 6 '12 at 21:04

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up vote 11 down vote accepted

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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Is there are more elementary way to see this though? –  198203 Mar 6 '12 at 20:47
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@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. –  Henning Makholm Mar 6 '12 at 20:54
    
How 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. –  Tyler Mar 6 '12 at 20:55
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Basically, 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. –  Thomas Andrews Mar 6 '12 at 22:24
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@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. –  Keenan Kidwell Mar 6 '12 at 22:53

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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How 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. –  André Nov 19 '12 at 6:43
    
If 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. –  checkmath Nov 23 '12 at 18:04

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