\ubca1\ud130\uacf5\uac04 \\(K^n,\\) \\(K^m\\) \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uacfc \\(m\\times n\\) \ud589\ub82c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n
\\(K\\)\uac00 \uccb4(field)\uc774\uace0 \\(n\\)\uacfc \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \uc131\ubd84\uc774 \\(K\\)\uc5d0 \uc18d\ud558\ub294 \\(m\\times n\\) \ud589\ub82c\ub4e4\uc758 \ubaa8\uc784\uc744\\(\\newcommand{\\MatK}{\\operatorname{Mat}_{m \\times n}(K)}\\)
\n\\[\\MatK\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \uc815\uc758\uc5ed\uc774 \\(K^n\\)\uc774\uace0 \uacf5\uc5ed\uc774 \\(K^m\\)\uc778 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ubaa8\uc784\uc744\\(\\newcommand{\\HomK}{\\operatorname{Hom}(K^n ,\\, K^m )}\\)
\n\\[\\HomK\\]
\n\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\uc5ec\uae30\uc11c \\(K^n\\)\uacfc \\(K^m\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \ubca1\ud130 \ud569\uacfc \uc2a4\uce7c\ub77c \uacf1\uc774 \uc8fc\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.]<\/p>\n
\uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \\(m\\times n\\) \ud589\ub82c \\(A = (a_{ij})_{m\\times n}\\), \\(B=(b_{ij})_{m\\times n}\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c\uacf1 \\(kA\\)\uc640 \ud569 \\(A+B\\)\ub97c \uac01\uac01
\n\\[\\begin{gather}
\nkA = (ka_{ij})_{m\\times n} ,\\\\[6pt]
\nA+B = (a_{ij} + b_{ij})_{m\\times n}
\n\\end{gather}\\]
\n\ub85c \uc815\uc758\ud558\uba74 \\(\\MatK\\)\ub294 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658 \\(T_1 ,\\) \\(T_2\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c\uacf1 \\(kT_1\\)\uacfc \ud569 \\(T_1 + T_2\\)\ub97c \uac01\uac01
\n\\[\\begin{gather}
\n\\forall \\mathbf{x}\\in K^n : \\, (kT_1) (\\mathbf{x}) = k(T_1 (\\mathbf{x})),\\\\[6pt]
\n\\forall \\mathbf{x}\\in K^n : \\, (T_1 + T_2 )(\\mathbf{x}) = T_1 (\\mathbf{x}) + T_2 (\\mathbf{x})
\n\\end{gather}
\n\\]
\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub85c \uc815\uc758\ud558\uba74 \\(\\HomK\\)\ub294 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\ub2e4.<\/p>\n
\uc774\uc81c \uc774 \ub450 \uacf5\uac04 \\(\\MatK\\)\uc640 \\(\\HomK\\)\uac00 \ub3d9\ud615\uc778 \ubca1\ud130\uacf5\uac04\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n
\uc815\ub9ac 1.<\/span> \ub450 \ubca1\ud130\uacf5\uac04\uc774 \ub3d9\ud615\uc784\uc744 \ubcf4\uc774\ub824\uba74 \ub450 \uacf5\uac04 \uc0ac\uc774\uc5d0 \uc815\uc758\ub41c \ub3d9\ud615\uc0ac\uc2f1\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \ubca1\ud130\uacf5\uac04\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub780 \uc77c\ub300\uc77c\ub300\uc751(one-to-one\uc774\uba74\uc11c onto)\uc778 \uc120\ud615\ubcc0\ud658(linear transformation)\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n \\(m\\times n\\) \ud589\ub82c \\(A\\)\uc5d0 \ub300\ud558\uc5ec \uc120\ud615\ubcc0\ud658 \\(T_A : K^n \\rightarrow K^m\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \uac01 \ud589\ub82c \\(A\\in\\MatK\\)\uc640 \\(n\\times 1\\) \ud589\ub82c \\(\\mathbf{x}\\)\uc758 \uacf1 \\(A\\mathbf{x}\\)\ub294 \ud589\ub82c\uc758 \uacf1\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc798 \uc815\uc758\ub418\uba70, \uacf1\ud55c \uacb0\uacfc\ub294 \\(m\\times 1\\) \ud589\ub82c\uc774 \ub41c\ub2e4. \ub610\ud55c \uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \ub450 \ubca1\ud130 \\(\\mathbf{x}_1 \\in K^n ,\\) \\(\\mathbf{x}_2 \\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \ud589\ub82c \\(A\\in\\MatK\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(T_{kA}\\)\uc640 \\(kT_A\\)\ub294 \ubaa8\ub450 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(\\mathbf{x}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc73c\ub85c \ub450 \ud589\ub82c \\(A,\\,B\\in\\MatK\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(T_{A+B}\\)\uc640 \\(T_A +T_B\\)\ub294 \ubaa8\ub450 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(\\mathbf{x}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc11c\ub85c \ub2e4\ub978 \ub450 \ud589\ub82c \\(A,\\,B\\in\\MatK\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(A\\)\uc640 \\(B\\)\uac00 \ub2e4\ub978 \ud589\ub82c\uc774\ubbc0\ub85c, \ub450 \ud589\ub82c\uc758 \uc5f4(column) \uc911\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(A\\)\uc640 \\(B\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc774 \uc11c\ub85c \ub2e4\ub974\ub2e4\uace0 \ud558\uc790. \uc989 \\(A^j \\ne B^j\\)\ub77c\uace0 \ud558\uc790. [\uc5ec\uae30\uc11c \uc704\ucca8\uc790 \\(j\\)\ub294 \uc81c\uacf1\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc774 \uc544\ub2c8\ub77c \\(j\\)\uc9f8 \uc5f4\uc744 \ub098\ud0c0\ub0b8\ub2e4.] \\(K^n\\)\uc758 \\(j\\)\uc9f8 \ud45c\uc900\uae30\uc800\uc6d0\uc18c\ub97c \\(\\mathbf{e}_j\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \uc120\ud615\ubcc0\ud658 \\(T\\in\\HomK\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(M(T)\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \uc774\uc81c \\(\\phi(M(T)) = T\\)\uc784\uc744 \ubcf4\uc774\uc790. \uac01 \\(j = 1,\\,2,\\,\\cdots,\\,n\\)\uc5d0 \ub300\ud558\uc5ec \uc774\ub85c\uc368 \uc784\uc758\uc758 \\(T\\in\\HomK\\)\uc5d0 \ub300\ud558\uc5ec \\(M(T)\\in\\MatK\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc704 \uc99d\uba85 \uacfc\uc815\uc758 5\ub2e8\uacc4\uc5d0\uc11c, \uc784\uc758\uc758 \\(T\\in\\HomK\\)\uc5d0 \ub300\ud558\uc5ec \\(T\\)\ub97c \ud45c\ud604\ud558\ub294 \ud589\ub82c \\(M(T)\\)\ub97c \ucc3e\uc558\uc73c\uba70, \uc774 \ud589\ub82c\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uc120\ud615\uc0ac\uc0c1\uc740 \\(T\\)\uc640 \uc77c\uce58\ud568\uc744 \ubcf4\uc600\ub2e4. \uc989 \uc784\uc758\uc758 \\(T\\in\\HomK\\)\uc5d0 \ub300\ud558\uc5ec \uc774 \uacfc\uc815\uc744 \ubc18\ub300\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc989 \uc784\uc758\uc758 \ud589\ub82c \\(A\\in\\MatK\\)\uc5d0 \ub300\ud558\uc5ec \\(A\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uc120\ud615\uc0ac\uc0c1 \\(T_A\\)\ub97c \ucc3e\uace0, \ub2e4\uc2dc \\(T_A\\)\uc758 \ud45c\ud604\ud589\ub82c \\(M(T_A)\\)\ub97c \ucc3e\uc744 \uc218 \uc788\ub2e4. \uc774 \ub450 \ud589\ub82c\uc774 \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 \\(A\\)\uac00 \\(m\\times n\\) \ud589\ub82c\uc774\ubbc0\ub85c \\(T_A\\)\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(M(T_A)\\)\ub294 \\(m\\times n\\) \ud589\ub82c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub450 \ud589\ub82c\uc758 \ud06c\uae30\ub294 \uc77c\uce58\ud55c\ub2e4. \ub2e4\uc74c\uc73c\ub85c \ub450 \ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \ube44\uad50\ud558\uba74 \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc694\uc57d\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(K^n,\\) \\(K^m\\) \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \\(m\\times n\\) \ud589\ub82c\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n \uc815\ub9ac 1\uc744 \uc99d\uba85\ud560 \ub54c \uc0ac\uc6a9\ud55c \ub3d9\ud615\uc0ac\uc0c1\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc720\uc6a9\ud55c \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4.<\/p>\n \n\uc815\ub9ac 2.<\/span> \uc99d\uba85<\/p>\n \\(AB\\)\uac00 \\(m\\times p\\) \ud589\ub82c\uc774\ubbc0\ub85c \\(T_{AB}\\)\ub294 \\(K^p\\)\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub610\ud55c \\(A\\)\uc640 \\(B\\)\uac00 \uac01\uac01 \\(m\\times n \\) \ud589\ub82c, \\(n\\times p\\) \ud589\ub82c\uc774\ubbc0\ub85c \\(T_A\\)\uc640 \\(T_B\\)\ub294 \uac01\uac01 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658, \\(K^p\\)\ub85c\ubd80\ud130 \\(K^n\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub530\ub77c\uc11c \\(T_A \\circ T_B\\)\ub294 \\(K^p\\)\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc989 \\(T_{AB}\\)\uc758 \uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc740 \\(T_A \\circ T_B\\)\uc758 \uc815\uc758\uc5ed, \uacf5\uc5ed\uacfc \uac19\ub2e4.<\/p>\n \uc774\uc81c \ub450 \ud568\uc218\uac00 \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc790. \uac01 \\(j=1,\\,2,\\,\\cdots,\\,p\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ub9ac 2\ub97c \uc0ac\uc6a9\ud558\uba74 \ud589\ub82c\uc758 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \uacb0\ud569\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\ub9ac 3.<\/span> \uc99d\uba85<\/p>\n \uba3c\uc800 \\((AB)C\\)\uc640 \\(A(BC)\\)\ub294 \ubaa8\ub450 \\(m\\times q\\) \ud589\ub82c\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(A,\\) \\(B,\\) \\(C\\)\uc5d0 \uc758\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \uc120\ud615\ubcc0\ud658\uc744 \\(T_A,\\) \\(T_B,\\) \\(T_C\\)\ub77c\uace0 \ud558\uba74 \uc815\ub9ac 2\uc758\uacb0\uacfc\uc640 \ud568\uc218\uc758 \ud569\uc131\uc758 \uacb0\ud569\ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \uc815\ub9ac 2\ub97c \ubc18\ub300 \ubc29\ud5a5\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n \n\uc815\ub9ac 4.<\/span> \uc99d\uba85<\/p>\n \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uc740 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c \\(T_2 \\circ T_1\\)\uc740 \\(K^p\\)\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub530\ub77c\uc11c \\(M(T_2 \\circ T_1)\\)\uc740 \\(m\\times p\\) \ud589\ub82c\uc774\ub2e4. \ub610\ud55c \\(M(T_2)\\)\uc640 \\(M(T_1)\\)\uc740 \uac01\uac01 \\(m\\times n\\) \ud589\ub82c, \\(n\\times p\\) \ud589\ub82c\uc774\ubbc0\ub85c, \ub450 \ud589\ub82c\uc758 \uacf1 \\(M(T_2)M(T_1)\\)\uc740 \\(m\\times p\\) \ud589\ub82c\uc774\ub2e4. \uc989 \\(M(T_2 \\circ T_1)\\)\uacfc \\(M(T_2)M(T_1)\\)\uc740 \ud06c\uae30\uac00 \uac19\uc740 \ud589\ub82c\uc774\ub2e4.<\/p>\n \uc774\uc81c \uc774 \ub450 \ud589\ub82c\uc758 \uc5f4\uc774 \ubaa8\ub450 \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc790. \uac01 \\(j=1,\\,2,\\,\\cdots,\\,p\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ub9ac 2, 3, 4\uc758 \ub0b4\uc6a9\uc744 \uc694\uc57d\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n \ud589\ub82c\uc758 \uacf1\uc740 \ud568\uc218\uc758 \ud569\uc131\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":" \ubca1\ud130\uacf5\uac04 \\(K^n,\\) \\(K^m\\) \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uacfc \\(m\\times n\\) \ud589\ub82c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(K\\)\uac00 \uccb4(field)\uc774\uace0 \\(n\\)\uacfc \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \uc131\ubd84\uc774 \\(K\\)\uc5d0 \uc18d\ud558\ub294 \\(m\\times n\\) \ud589\ub82c\ub4e4\uc758 \ubaa8\uc784\uc744\\(\\newcommand{\\MatK}{\\operatorname{Mat}_{m \\times n}(K)}\\) \\(\\MatK\\) \ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \uc815\uc758\uc5ed\uc774 \\(K^n\\)\uc774\uace0 \uacf5\uc5ed\uc774 \\(K^m\\)\uc778 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ubaa8\uc784\uc744\\(\\newcommand{\\HomK}{\\operatorname{Hom}(K^n ,\\, K^m )}\\) \\(\\HomK\\) \uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\uc5ec\uae30\uc11c \\(K^n\\)\uacfc \\(K^m\\)\uc740 \ud1b5\uc0c1\uc801\uc778 \ubca1\ud130 \ud569\uacfc \uc2a4\uce7c\ub77c \uacf1\uc774 \uc8fc\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.] \uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \\(m\\times n\\) \ud589\ub82c \\(A = (a_{ij})_{m\\times n}\\),…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[57],"tags":[544,429,428,458],"class_list":["post-6225","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-544","tag-429","tag-428","tag-458"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6225","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6225"}],"version-history":[{"count":55,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6225\/revisions"}],"predecessor-version":[{"id":6314,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6225\/revisions\/6314"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=6225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=6225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\ub450 \ubca1\ud130\uacf5\uac04 \\(\\MatK\\)\uc640 \\(\\HomK\\)\uc740 \ub3d9\ud615\uc774\ub2e4.\n<\/p>\n<\/div>\n1\ub2e8\uacc4. \ub450 \uacf5\uac04 \uc0ac\uc774\uc758 \ud568\uc218 \\(\\phi\\)\ub97c \uc815\uc758\ud558\uc790.<\/h4>\n
\n\\[\\forall \\mathbf{x} \\in K^n :\\, T_A(\\mathbf{x}) = A\\mathbf{x}.\\]
\n\uc989 \\(T_A\\)\ub294 \\(K^n\\)\uc758 \ubca1\ud130 \\(\\mathbf{x}\\)\uc758 \uc67c\ucabd\uc5d0 \ud589\ub82c \\(A\\)\ub97c \uacf1\ud55c \uacb0\uacfc\ub97c \ud568\uc22b\uac12\uc73c\ub85c \ucde8\ud558\ub294 \ud568\uc218\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\mathbf{x}\\)\ub294 \\(K^n\\)\uc758 \uc6d0\uc18c(\ubca1\ud130)\ub97c \\(n\\times 1\\) \ud589\ub82c\ub85c \ub098\ud0c0\ub0b8 \uc5f4\ubca1\ud130\uc774\ub2e4. \uc774\uc81c \ud568\uc218
\n\\[\\phi : \\MatK \\rightarrow \\HomK\\]
\n\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[\\forall A \\in \\MatK : \\, \\phi (A) = T_A .\\]
\n\uc774 \ud568\uc218\uac00 \ubc14\ub85c \uc6b0\ub9ac\uac00 \ubc14\ub77c\ub294 \ub3d9\ud615\uc0ac\uc0c1\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n2\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/h4>\n
\n\\[\\begin{gather}
\nT_A (k\\mathbf{x}_1 ) = A(k\\mathbf{x}_1) = k(A\\mathbf{x}_1) = k T_A (\\mathbf{x}_1 ), \\\\[6pt]
\nT_A (\\mathbf{x}_1 + \\mathbf{x}_2) = A(\\mathbf{x}_1 + \\mathbf{x}_2 ) = A\\mathbf{x}_1 + A\\mathbf{x}_2 = T_A (\\mathbf{x}_1) + T_A(\\mathbf{x}_2 )
\n\\end{gather}\\]
\n\uc774\ubbc0\ub85c \\(T_A\\)\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc989 \\(\\phi\\)\ub294 \\(\\MatK\\)\uc758 \uc6d0\uc18c\ub97c \\(\\HomK\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\phi\\)\ub294 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/p>\n3\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub2e4.<\/h4>\n
\n\\[T_{kA} (\\mathbf{x}) = (kA)\\mathbf{x} = k(A\\mathbf{x}) = k(T_A(\\mathbf{X})) = (kT_A)(\\mathbf{x})\\]
\n\uc774\ub2e4. \uc989 \uc815\uc758\uc5ed \\(K^n\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \\(T_{kA}\\)\uc640 \\(kT_A\\)\uc758 \ud568\uc22b\uac12\uc774 \uc77c\uce58\ud558\ubbc0\ub85c, \uc774 \ub450 \ud568\uc218\ub294 \uac19\uc740 \ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\phi(kA) = T_{kA} = kT_A = k\\phi(A)\\]
\n\uc774\ub2e4.<\/p>\n
\n\\[T_{A+B}(\\mathbf{x}) = (A+B)\\mathbf{x} = A\\mathbf{x}+B\\mathbf{x} = T_A (\\mathbf{x}) + T_B (\\mathbf{x}) = (T_A + T_B)(\\mathbf{x})\\]
\n\uc774\ub2e4. \uc989 \uc815\uc758\uc5ed \\(K^n\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \\(T_{A+B}\\)\uc640 \\(T_A +T_B\\)\uc758 \ud568\uc22b\uac12\uc774 \uc77c\uce58\ud558\ubbc0\ub85c, \uc774 \ub450 \ud568\uc218\ub294 \uac19\uc740 \ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\phi(A+B) = T_{A+B} = T_A + T_B = \\phi(A) + \\phi(B)\\]
\n\uc774\ub2e4. \uc774\ub85c\uc368 \\(\\phi\\)\uac00 \uc120\ud615\uc784\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n4\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218(one-to-one function)\uc774\ub2e4.<\/h4>\n
\n\\[T_A (\\mathbf{e}_j) = A\\mathbf{e}_j = A^j \\ne B^j = B\\mathbf{e}_j = T_B(\\mathbf{e}_j)\\]
\n\uc774\ub2e4. \uc989 \uc815\uc758\uc5ed \\(K^n\\)\uc758 \uc6d0\uc18c \\(\\mathbf{e}_j\\)\uc5d0 \ub300\ud558\uc5ec \\(T_A\\)\uc640 \\(T_B\\)\uc758 \ud568\uc22b\uac12\uc774 \ub2e4\ub974\ubbc0\ub85c \ub450 \ud568\uc218\ub294 \ub2e4\ub978 \ud568\uc218\uc774\ub2e4. \uc774\ub85c\uc368 \\(A\\ne B\\)\uc77c \ub54c
\n\\[\\phi(A) = T_A \\ne T_B = \\phi(B)\\]
\n\uc774\ubbc0\ub85c \\(\\phi\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/p>\n5\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218(onto)\uc774\ub2e4.<\/h4>\n
\n\\[M(T) = [ T(\\mathbf{e}_1) \\,\\,\\, T(\\mathbf{e}_2) \\,\\,\\, \\cdots \\,\\,\\, T(\\mathbf{e}_n)]\\]
\n\uc989 \\(M(T)\\)\ub294 \\(j\\)\uc9f8 \uc5f4\uc774 \\(T(\\mathbf{e}_j)\\)\uc640 \uc77c\uce58\ud558\ub294 \ud589\ub82c\uc774\ub2e4. \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(T(\\mathbf{e}_j)\\)\ub294 \\(m\\times 1\\) \ud589\ub82c\uc774\ubbc0\ub85c \\(M(T)\\)\ub294 \\(m\\times n\\) \ud589\ub82c\uc774\ub2e4. \uc989 \\(M(T)\\in\\MatK\\)\uc774\ub2e4. \uc774\ub54c \\(M(T)\\)\ub97c \uc67c\ucabd\uc5d0 \uacf1\ud558\ub294 \uc5f0\uc0b0\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(T_{M(T)}\\)\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4.<\/p>\n
\n\\[T_{M(T)}(\\mathbf{e}_j) = (M(T))\\mathbf{e}_j = (M(T))^j\\]
\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\((M(T))^j\\)\ub294 \ud589\ub82c \\(M(T)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \ub098\ud0c0\ub0b4\uba70, \uc774\uac83\uc740 \\(T(\\mathbf{e}_j)\\)\uc640 \uac19\ub2e4. \uc989
\n\\[T_{M(T)}(\\mathbf{e}_j) = (M(T))\\mathbf{e}_j = (M(T))^j = T(\\mathbf{e}_j)\\]
\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\mathbf{e}_1 ,\\) \\(\\mathbf{e}_2 ,\\) \\(\\cdots,\\) \\(\\mathbf{e}_n \\)\uc740 \\(K^n\\)\uc758 \uae30\uc800\uc774\ubbc0\ub85c, \uc774 \ubca1\ud130\ub4e4\uc758 \ud568\uc22b\uac12\uc774 \uc77c\uce58\ud558\ub294 \ub450 \uc120\ud615\ubcc0\ud658\uc740 \uc644\uc804\ud788 \uc77c\uce58\ud55c\ub2e4. \uc989 \\(T_{M(T)}\\)\uc640 \\(T\\)\ub294 \uac19\uc740 \ud568\uc218\uc774\ub2e4.<\/p>\n
\n\\[\\phi(M(T)) = T_{M(T)} = T\\]
\n\uc774\ubbc0\ub85c \\(\\phi\\)\ub294 \\(\\HomK\\) \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.
\n<\/span><\/p>\n
\n\\[T_{M(T)} = T\\]
\n\uc774\ub2e4.<\/p>\n
\n\\[(M(T_A))^j = T_A (\\mathbf{e}_j) = A\\mathbf{e}_j = A^j\\]
\n\uc774\ubbc0\ub85c, \ub450 \ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(A\\in\\MatK\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[M(T_A) = A\\]
\n\uc774\ub2e4. \uc774\ub85c\uc368 \\(\\MatK\\)\uc758 \ud589\ub82c \\(A\\)\uc640 \\(\\HomK\\)\uc758 \ud589\ub82c \\(T_A\\)\ub294 \ud558\ub098\uc529 \ub300\uc751\ub418\uba70, \uc774 \ub300\uc751\uc740 \ub450 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n
\n\\(m,\\) \\(n,\\) \\(p\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\uace0
\n\\(\\newcommand{\\MatKp}{\\operatorname{Mat}_{n \\times p}(K)}\\)
\n\\(\\newcommand{\\HomKp}{\\operatorname{Hom}(K^p ,\\, K^n)}\\)
\n\\[A \\in \\MatK ,\\,\\, B \\in \\MatKp\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[T_{AB} = T_A \\circ T_B\\]
\n\uc774\ub2e4. \uc989 \ub450 \ud589\ub82c\uc744 \uacf1\ud55c \ud589\ub82c\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4 \uc120\ud615\ubcc0\ud658\uc740 \uac01 \ud589\ub82c\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4 \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n
\n\\[\\begin{align}
\nT_{AB}(\\mathbf{e}_j)
\n&= (AB)\\mathbf{e}_j \\\\[6pt]
\n&= (AB)^j \\\\[6pt]
\n&= A\\cdot B^j \\\\[6pt]
\n&= T_A(B^j) \\\\[6pt]
\n&= T_A(B\\mathbf{e}_j) \\\\[6pt]
\n&= T_A(T_B(\\mathbf{e}_j)) \\\\[6pt]
\n&= (T_A\\circ T_B)(\\mathbf{e}_j)
\n\\end{align}\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \uc704\ucca8\uc790 \\(j\\)\ub294 \ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \ub098\ud0c0\ub0b4\uba70 \\(\\mathbf{e}_j\\)\ub294 \\(K^p\\)\uc758 \ud45c\uc900\uae30\uc800\uc6d0\uc18c \uc911 \\(j\\)\uc9f8 \ubca1\ud130\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uc815\uc758\uc5ed\uc758 \uc784\uc758\uc758 \uae30\uc800\uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \ub450 \ud568\uc218 \\(T_{AB}\\)\uc640 \\(T_A \\circ T_B\\)\uc758 \uac12\uc774 \uc77c\uce58\ud558\ubbc0\ub85c, \ub450 \ud568\uc218\ub294 \uac19\uc740 \ud568\uc218\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n
\n\\(m,\\) \\(n,\\) \\(p,\\) \\(q\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\uace0
\n\\[
\nA\\in\\operatorname{Mat}_{m\\times n}(K),\\,\\,
\nB\\in\\operatorname{Mat}_{n\\times p}(K),\\,\\,
\nC\\in\\operatorname{Mat}_{p\\times q}(K)
\n\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[(AB)C = A(BC)\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n
\n\\[\\begin{align}
\nT_{(AB)C} &= T_{AB}\\circ T_C \\\\[6pt]
\n&= (T_A \\circ T_B)\\circ T_C \\\\[6pt]
\n&= T_A \\circ (T_B \\circ T_C) \\\\[6pt]
\n&= T_A \\circ T_{BC} \\\\[6pt]
\n&= T_{A(BC)}
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[(AB)C = M(T_{(AB)C}) = M(T_{A(BC)})=A(BC)\\]
\n\uc774\ub2e4.
\n<\/p>\n<\/div>\n
\n\\(m,\\) \\(n,\\) \\(p\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\uace0
\n\\(\\newcommand{\\MatKp}{\\operatorname{Mat}_{n \\times p}(K)}\\)
\n\\(\\newcommand{\\HomKp}{\\operatorname{Hom}(K^p ,\\, K^n)}\\)
\n\\[T_1 \\in \\HomKp ,\\,\\, T_2 \\in \\HomK\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[M(T_2 \\circ T_1) = M(T_2 ) M(T_1)\\]
\n\uc774\ub2e4. \uc989 \ub450 \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uc740 \uac01 \ud45c\ud604\ud589\ub82c\uc744 \uacf1\ud55c \uac83\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n
\n\\[\\begin{align}
\n(M(T_2 \\circ T_1))^j
\n&= (T_2 \\circ T_1)(\\mathbf{e}_j) \\\\[6pt]
\n&= T_2(T_1(\\mathbf{e}_j)) \\\\[6pt]
\n&= T_2 (M(T_1)^j) \\\\[6pt]
\n&= T_2 ((M(T_1))\\mathbf{e}_j) \\\\[6pt]
\n&= (M(T_2))((M(T_1))\\mathbf{e}_j) \\\\[6pt]
\n&= (M(T_2)M(T_1))(\\mathbf{e}_j) \\\\[6pt]
\n&= (M(T_2)M(T_1))^j
\n\\end{align}\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \uc704\ucca8\uc790 \\(j\\)\ub294 \ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \ub098\ud0c0\ub0b4\uba70 \\(\\mathbf{e}_j\\)\ub294 \\(K^p\\)\uc758 \ud45c\uc900\uae30\uc800\uc6d0\uc18c \uc911 \\(j\\)\uc9f8 \ubca1\ud130\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ub450 \ud589\ub82c \\(M(T_2 \\circ T_1)\\)\uacfc \\(M(T_2)M(T_1)\\)\uc758 \ubaa8\ub4e0 \uc5f4\uc774 \uc77c\uce58\ud558\ubbc0\ub85c, \ub450 \ud589\ub82c\uc740 \uac19\uc740 \ud589\ub82c\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n