General Linear Groups — Generally « The Unapologetic Mathematician

General Linear Groups — Generally

Monday, we saw that the general linear groups $\mathrm{GL}_n(\mathbb{F})$ are matrix groups, specifically consisting of those whose columns are linearly independent. But what about more general vector spaces?

Well, we know that every finite-dimensional vector space has a basis, and is thus isomorphic to $\mathbb{F}^n$ , where $n$ is the cardinality of the basis. So given a vector space $V$ with a basis $\{f_i\}$ of cardinality $n$ , we have the isomorphism $S:\mathbb{F}^n\rightarrow V$ defined by $S(e_i)=f_i$ and $S^{-1}(f_i)=e_i$ .

This isomorphism of vector spaces then induces an isomorphism of their automorphism groups. That is, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$ . Given an invertible linear transformation $T:V\rightarrow V$ , we can conjugate it by $S$ to get $S^{-1}TS:\mathbb{F}^n\rightarrow\mathbb{F}^n$ . This has inverse $S^{-1}T^{-1}S$ , and so is an element of $\mathrm{GL}_n(\mathbb{F})$ . Thus (not unexpectedly) every invertible linear transformation from a vector space $V$ to itself gets an invertible matrix.

But this assignment depends essentially on the arbitrary choice of the basis $\{f_i\}$ for $V$ . What if we choose a different basis $\{\tilde{f}_i\}$ ? Then we get a new transformation $\tilde{S}$ and a new isomorphism of groups $T\mapsto\tilde{S}^{-1}T\tilde{S}$ . But this gives us an inner automorphism of $\mathrm{GL}_n(\mathbb{F})$ . Given a transformation $M:\mathbb{F}^n\rightarrow\mathbb{F}^n$ , we get the transformation
$\tilde{S}^{-1}SMS^{-1}\tilde{S}=\left(\tilde{S}^{-1}S\right)^{-1}M\left(\tilde{S}^{-1}S\right):\mathbb{F}^n\rightarrow\mathbb{F}^n$
This composite $\tilde{S}^{-1}S$ sends $\mathbb{F}^n$ to itself, and it has an inverse. Thus changing the basis on $V$ induces an inner automorphism of the matrix group $\mathrm{GL}_n(\mathbb{F})$ .

Now let’s consider a linear transformation $T:V\rightarrow V$ . We have two bases for $V$ , and thus two different matrices — two different elements of $\mathrm{GL}_n(\mathbb{F})$ — corresponding to $T$ : $S^{-1}TS$ and $\tilde{S}^{-1}T\tilde{S}$ . We get from one to the other by conjugation with $\tilde{S}^{-1}S$ :

$\left(\tilde{S}^{-1}S\right)S^{-1}TS\left(\tilde{S}^{-1}S\right)^{-1}=\tilde{S}^{-1}SS^{-1}TSS^{-1}\tilde{S}=\tilde{S}^{-1}T\tilde{S}$

And what is this transformation $\tilde{S}^{-1}S$ ? How does it act on a basis vector in $\mathbb{F}^n$ ? We calculate:
$\tilde{S}^{-1}(S(e_j))=\tilde{S}^{-1}(f_j)=\tilde{S}^{-1}(x_j^i\tilde{f}_i)=x_j^i\tilde{S}^{-1}(\tilde{f}_i)=x_j^ie_i$
where $f_j=x_j^i\tilde{f}_i$ expresses the vectors in one basis for $V$ in terms of those of the other. That is, the $j$ th column of the matrix $X$ consists of the components of $f_j$ written in terms of the $\tilde{f}_i$ . Similarly, the inverse matrix $X^{-1}$ with entries $\tilde{x}_i^j$ , writes the $\tilde{f}_j$ in terms of the $f_i$ : $\tilde{f}_i=\tilde{x}_i^jf_j$ .

It is these “change-of-basis” matrices that effect all of our, well, changes of basis. For example, say we have a vector $v\in V$ with components $v=v^jf_j$ . Then we can expand this:

$v=v^jf_j=v^k\delta_k^jf_j=v^kx_k^i\tilde{x}_i^jf_j=\left(x_k^iv^k\right)\tilde{f}_i$

So our components in the new basis are $\tilde{v}^i=x_k^iv^k$ .

As another example, say that we have a linear transformation $T:V\rightarrow V$ with matrix components $t_i^j$ with respect to the basis $\{f_i\}$ . That is, $T(f_i)=t_i^jf_j$ . Then we can calculate:

$T(\tilde{f}_i)=T(\tilde{x}_i^kf_k)=\tilde{x}_i^kT(f_k)=\tilde{x}_i^kt_k^lf_l=\tilde{x}_i^kt_k^lx_l^j\tilde{f}_j$

and we have the new matrix components $\tilde{t}_i^j=\tilde{x}_i^kt_k^lx_l^j$ .

October 22, 2008 - Posted by John Armstrong | Algebra, Group Examples, Linear Algebra

12 Comments »

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