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"quote": "复数情况也可作类似讨论：全体 $n\\times n$ 非奇异复矩阵组成群 $GL(n,\\mathbb{C})$。除 $n=1$ 外，这些群均非阿贝尔，因为一般有 $\\mathsf{A}\\mathsf{B}\\ne\\mathsf{B}\\mathsf{A}$。它们并称为$n$ 阶一般线性群(general linear group)。以矩阵为元素、矩阵乘法为组成法则的子群统称为矩阵群(matrix group)。",
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"quote": "读者应注意，本书常把线性算符 $L$ 与它在某个未指定基下的矩阵（矩阵元记为 $L_{jk}$）使用同一个符号。这是因为我们通常关心算符及其矩阵表示中与基无关的性质，例如它们是否可逆。$V$ 上所有可逆[线性算符(linear operators)](../../../../../concepts/linear-algebra/linear-map)在复合下构成一个群，记作 $GL(V)$；其中“GL”取自英文“General Linear”（一般线性）；这个群称为[一般线性群(general linear group)](../../../../../concepts/linear-algebra/general-linear-group)。选定一组基后，该群即可与可逆矩阵群对应，群运算就是矩阵乘法。若 $\\dim V=n$，则实数域与复数域上的群分别记作 $GL(n,\\mathbf R)$ 与 $GL(n,\\mathbf C)$。",
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"quote": "> [!recall]- 回顾：一般线性群 → [原定义](../chapter-04/01-vector-spaces-and-linear-maps#concept-4-general-linear-group) \n> - [一般线性群(general linear group)](../../../../../concepts/linear-algebra/general-linear-group) $GL(n,\\mathbf{R})$ 和 $GL(n,\\mathbf{C})$ 分别是所有可逆实矩阵和复矩阵构成的群。它们的李代数分别是 $\\mathfrak{gl}(n,\\mathbf{R})=M(n,\\mathbf{R})$ 和 $\\mathfrak{gl}(n,\\mathbf{C})=M(n,\\mathbf{C})$。这两个向量空间由所有 $n\\times n$ 矩阵构成，李括号为矩阵对易子。其他李群是这些群的子群，相应的李代数是这些矩阵李代数的李子代数。\n>\n> ",
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"quote": "> [!recall]- Recall: general linear group → [original definition](../chapter-04/01-vector-spaces-and-linear-maps#concept-4-general-linear-group) \n> - The [general linear groups](../../../../../concepts/linear-algebra/general-linear-group) $GL(n,\\mathbf{R})$ and $GL(n,\\mathbf{C})$ are the groups of all invertible real and complex matrices, respectively. Their Lie algebras are $\\mathfrak{gl}(n,\\mathbf{R})=M(n,\\mathbf{R})$ and $\\mathfrak{gl}(n,\\mathbf{C})=M(n,\\mathbf{C})$. These are the vector spaces of all $n\\times n$ matrices, with Lie bracket given by the matrix commutator. Other Lie groups are subgroups of these groups, with Lie algebras that are sub-Lie algebras of the corresponding matrix Lie algebras.\n>\n> ",
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