Franklin's Notes

Matrix groups and linear groups

Given a field $\mathbb k$, the set of full-rank $n\times n$ matrices over $\mathbb k$ constitute a group whose group operation is matrix multiplication. This group is called the general linear group of dimension $n$ over $\mathbb k$, and is denoted $\mathrm{GL}_n(\mathbb k)$. In general, a matrix group is a subgroup of $\mathrm{GL}_n(\mathbb k)$, and a linear group is any group $G$ that is isomorphic to a matrix group. Here's a list of common examples of matrix groups:

Given a polynomial or a form $f\in \mathbb k[x]$ in $n$ variables, we may also consider the subgroup of $A\in \mathrm{GL}_n(\mathbb k)$ satisfying $f(\vec x)=f(A\vec x)$. This is the stabilizer with respect to $f$ of the pre-composition group action of $\mathrm{GL}_n(\mathbb k)$ on $\mathbb k[x]$.





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