1087: General Linear Group of Finite-Dimensional Real Vectors Space

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definition of general linear group of finite-dimensional real vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of general linear group of vectors space.
• The reader knows a definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field.
• The reader knows a definition of canonical topology for finite-dimensional real vectors space.
• The reader knows a definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space.
• The reader knows a definition of open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that from any module with any basis into any module, a linear map can be defined by mapping the basis and linearly expanding the mapping.

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Target Context

• The reader will have a definition of general linear group of finite-dimensional real vectors space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$\mathbb{R}$: $\in \{\text{ the fields }\}$
$V$: $\in \{\text{ the }d\text{ -dimensional }\mathbb{R}\text{ vectors spaces }\}$
$\ast GL(V)$: $=\text{ the general linear group of }V$, $\in \{\text{ the }{d}^2\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
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Conditions:
$GL(V)$ is the open submanifold of the $C^{\mathrm{\infty }}$ manifold, $L(V:V)$, with the canonical topology and the canonical $C^{\mathrm{\infty }}$ atlas
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2: Note

The reason why this special case of general linear group of vectors space is taken up specifically is that this concept is also a $C^{\mathrm{\infty }}$ manifold: the field needs to be $\mathbb{R}$ and $V$ needs to be finite-dimensional for this concept.

Let us see that $GL(V)$ is indeed an open submanifold of $L(V:V)$.

Let us see that $L(V:V)$ is indeed a $d^2$-dimensional $\mathbb{R}$ vectors space.

Note for the definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field has already shown that it is an $\mathbb{R}$ vectors space.

Let any basis of $V$ be $\{b_1,...,b_d\}$.

Let $b_k^j\in L(V:V)$ be the one that maps $b_j$ to $b_k$ and maps any other basis element to $0$, which is indeed in $L(V:V)$, by the proposition that from any module with any basis into any module, a linear map can be defined by mapping the basis and linearly expanding the mapping.

Let us see that $\{b_k^j|j\in \{1,...,d\},k\in \{1,...,d\}\}$ is a basis for $L(V:V)$.

Let us suppose that $v_j^k{b}_k^j=0$ for any $v_j^k\in \mathbb{R}$. Let it operate on $b_l$. $v_j^k{b}_k^j(b_l)=0(b_l)=0$, but $v_j^k{b}_k^j(b_l)=v_l^k{b}_k^l(b_l)=v_l^k{b}_k$, which implies that $v_l^k=0$. So, $\{b_k^j\}$ is linearly independent.

For each fixed $j$, $v_l^k{b}_k^l(b_j)=v_j^k{b}_k$ spans $V$ by choosing $v_j^k$ s. So, each element of $L(V:V)$ is expressed as a $v_l^k{b}_k^l$.

So, $\{b_k^j\}$ is a basis for $L(V:V)$, and so, $L(V:V)$ is $d^2$-dimensional.

So, $L(V:V)$ can be made a $C^{\mathrm{\infty }}$ manifold, by the definition of canonical topology for finite-dimensional real vectors space and the definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space.

By the definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space, $(L(V:V)\subseteq L(V:V),\varphi )$ where $\varphi :L(V:V)\to {\mathbb{R}}^{d^2},v=v_j^k{b}_k^j\mapsto M(v)=\left(\begin{array}{ccc}{v}_1^1& ...& v_d^1\\ ...\\ v_1^d& ...& v_d^d\end{array}\right)$ is a global chart.

$\varphi (GL(V)):=S=\{M(v)\in {\mathbb{R}}^{d^2}|detM(v)\ne 0\}={\mathbb{R}}^{d^2}\setminus \{M(v)\in {\mathbb{R}}^{d^2}|detM(v)=0\}$.

But $\{M(v)\in {\mathbb{R}}^{d^2}|detM(v)=0\}$ is a closed subset of ${\mathbb{R}}^{d^2}$, because $det:{\mathbb{R}}^{d^2}\to \mathbb{R}$ is continuous and $\{M(v)\in {\mathbb{R}}^{d^2}|detM(v)=0\}=de{t}^{-1}(\{0\})$, where $\{0\}\in \mathbb{R}$ is a closed subset.

So, $S$ is an open subset of ${\mathbb{R}}^{d^2}$. $GL(V)={\varphi }^{-1}(S)$ is an open subset of $L(V:V)$.

So, $GL(V)$ is an open submanifold of $L(V:V)$.

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References

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