{"id":5497,"date":"2020-10-15T12:32:12","date_gmt":"2020-10-15T03:32:12","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5497"},"modified":"2020-11-05T21:51:25","modified_gmt":"2020-11-05T12:51:25","slug":"linear-algebra-representation-of-linear-transformations","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-representation-of-linear-transformations\/","title":{"rendered":"\uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\uacfc \uae30\uc800\uc758 \ubcc0\ud658"},"content":{"rendered":"<p>\uccb4 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \ucc28\uc6d0\uc774 \\(n\\)\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc740 \\(K^n\\)\uc640 \ub3d9\ud615\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \uc801\ub2f9\ud55c \\(n,\\) \\(m\\)\uc5d0 \ub300\ud558\uc5ec \uc120\ud615\ubcc0\ud658 \\(T : K^n \\rightarrow K^m\\)\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \ub354\uc6b1\uc774 \\(T : K^n \\rightarrow K^m\\)\uc740 \ud589\ub82c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\ubbc0\ub85c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \ud589\ub82c\uacfc \ub3d9\uc77c\uc2dc\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7f0\ub370 \uc774\ub7ec\ud55c \ud589\ub82c \ud45c\ud604\uc740 \ubca1\ud130\uacf5\uac04\uc5d0 \uc5b4\ub5a0\ud55c \uae30\uc800\uac00 \uc8fc\uc5b4\uc84c\ub294\uc9c0\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4.<\/p>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}<br \/>\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}<br \/>\n\\]\n<\/div>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc744 \ud589\ub82c\ub85c \ub098\ud0c0\ub0b4\uace0, \uae30\uc800\uac00 \ubc14\ub00c\uc5c8\uc744 \ub54c \uc120\ud615\ubcc0\ud658\uc744 \ud45c\ud604\ud558\ub294 \ud589\ub82c\uc774 \uc5b4\ub5bb\uac8c \ubc14\ub00c\ub294\uc9c0 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\uc120\ud615\ubcc0\ud658\uc758 \uacf5\uac04<\/h2>\n<p>\\(V\\)\uc640 \\(W\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(V\\)\ub85c\ubd80\ud130 \\(W\\)\ub85c\uc758 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ubaa8\uc784\uc744 \\(\\Hom(V,\\,W)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(f,\\,g\\in\\Hom(V,\\,W)\\)\uc640 \\(\\lambda\\in K\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lambda f\\)\uc640 \\(f+g\\)\ub97c<br \/>\n\\[\\begin{align}<br \/>\n(\\lambda f)(v) &#038;= \\lambda (f(v)) \\,\\text{ for all } v\\in V ,\\\\[5pt]<br \/>\n(f+g)(v) &#038;= f(v) + g(v) \\,\\text{ for all } v\\in V<br \/>\n\\end{align}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub85c \uc815\uc758\ud558\uba74, \uc774\ub7ec\ud55c \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(\\Hom(V,\\,W)\\)\ub294 \ubca1\ud130\uacf5\uac04\uc774 \ub41c\ub2e4.<\/p>\n<p>\\(U,\\) \\(V,\\) \\(W\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(g\\in\\Hom (V,\\,W),\\) \\(f_1 ,\\,f_2 \\in \\Hom (U,\\,V)\\)\uc5d0 \ub300\ud558\uc5ec\\[g\\circ (f_1 + f_2 ) =g\\circ f_1 + g\\circ f_2 .\\]<\/li>\n<li>\\(g_1 ,\\, g_2 \\in\\Hom(V,\\,W),\\) \\(f\\in\\Hom (U,\\,V)\\)\uc5d0 \ub300\ud558\uc5ec\\[(g_1 + g_2 ) \\circ f = g_1 \\circ f + g_2 \\circ f .\\]<\/li>\n<li>\\(g\\in\\Hom (V,\\,W),\\) \\(f\\in\\Hom(U,\\,V),\\) \\(\\lambda\\in K\\)\uc5d0 \ub300\ud558\uc5ec\\[(\\lambda g)\\circ f = g\\circ (\\lambda f) = \\lambda (g\\circ f).\\]<\/li>\n<\/ul>\n<p>\ud2b9\ud788 \\(f,\\,g\\in\\Hom(V,\\,V)\\)\uc640 \\(\\lambda\\in K\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[(\\lambda g) \\circ f = g\\circ (\\lambda f) = \\lambda (g\\circ f).\\tag{1.1}\\]<br \/>\n\\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc5d0 \ud658\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uacf1 \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc838 \uc788\uace0 \uc2dd (1.1)\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \uadf8\ub7ec\ud55c \uacf5\uac04\uc744 <span class=\"defined\">K-\ub300\uc218<\/span>(K-algebra)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \\(\\Hom(V,\\,V)\\)\ub294 \uc2a4\uce7c\ub77c\uacf1, \ubca1\ud130\ud569, \uacf1 \uc5f0\uc0b0(\ud568\uc218\ud569\uc131)\uc774 \uc8fc\uc5b4\uc9c4 K-\ub300\uc218\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span><br \/>\n\\(f,\\,g\\in \\Hom(V,\\,W)\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(S\\)\uac00 \\(V\\)\ub97c \uc0dd\uc131\ud558\ub294 \uc9d1\ud569\uc774\uace0, \\(S\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc640 \\(g\\)\uc758 \ud568\uc22b\uac12\uc774 \uc77c\uce58\ud558\uba74 \\(f\\)\uc640 \\(g\\)\ub294 \uac19\uc740 \ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(v\\in V\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(S\\)\uc758 \uc6d0\uc18c \\(v_1 ,\\) \\(v_2 ,\\) \\(\\cdots ,\\) \\(v_n\\)\uacfc \uc2a4\uce7c\ub77c \\(\\lambda_j\\)\ub4e4\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[v = \\sum_{j=1}^n \\lambda_j v_j\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f\\)\uc640 \\(g\\)\uac00 \uc120\ud615\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nf(v) &#038;= f \\left( \\sum_{j=1}^n \\lambda_j v_j \\right)<br \/>\n= \\sum_{j=1}^n \\lambda_j f( v_j) \\\\[5pt]<br \/>\n&#038;= \\sum_{j=1}^n \\lambda_j g( v_j)<br \/>\n= g \\left( \\sum_{j=1}^n \\lambda_j v_j \\right) = g(v).<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(V\\) \uc704\uc5d0\uc11c \\(f=g\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\\(\\Hom(K^n ,\\, K^m )\\)\uc758 \ud45c\ud604<\/h2>\n<p>\uc774\uc81c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uacfc \ud589\ub82c \uc0ac\uc774\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\\(T \\in\\Hom (K^n ,\\, K^m )\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ud589\ub82c \\(M(T) \\in \\Mat_{m\\times n} (K)\\)\ub97c \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uac00 \\(T(\\mathbf{e}_j )\\)\uc778 \ud589\ub82c\ub85c \uc815\uc758\ud55c\ub2e4. \ud589\ub82c \\(M(T)\\)\ub97c <span class=\"defined\">\ud45c\uc900\uae30\uc800\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658 \\(T : \\mathbb{R}^3 \\rightarrow \\mathbb{R}^2\\)\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[T : \\left(\\begin{array}{} x_1 \\\\ x_2 \\\\ x_3 \\end{array}\\right) \\mapsto \\left(\\begin{array}{} 2x_1 +x_3 \\\\ x_2 &#8211; x_3 \\end{array}\\right).\\tag{2.1}\\]<br \/>\n\uadf8\ub7ec\uba74<br \/>\n\\[\\begin{align}<br \/>\nT(\\mathbf{e}_1) &#038;= \\left(\\begin{array}{} 2 \\\\ 0 \\end{array}\\right), \\\\[5pt]<br \/>\nT(\\mathbf{e}_2) &#038;= \\left(\\begin{array}{} 0 \\\\ 1 \\end{array}\\right), \\\\[5pt]<br \/>\nT(\\mathbf{e}_3) &#038;= \\left(\\begin{array}{r} 1 \\\\ -1 \\end{array}\\right)<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[M(T) = \\left(\\begin{array}{ccr} 2 &#038; 0 &#038; 1 \\\\ 0 &#038; 1 &#038; -1 \\end{array}\\right).\\tag{2.2}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc704 \uc608\uc81c\uc5d0\uc11c\ub294 (2.1)\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc73c\ub85c\ubd80\ud130 (2.2)\uc758 \ud589\ub82c\uc744 \uc5bb\uc5c8\ub2e4. \uc5ed\uc73c\ub85c (2.2)\uc640 \uac19\uc740 \ud589\ub82c\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uba74<br \/>\n\\[<br \/>\nT : \\left(\\begin{array}{} x_1 \\\\ x_2 \\\\ x_3 \\end{array}\\right) \\mapsto<br \/>\n\\left(\\begin{array}{ccr} 2 &#038; 0 &#038; 1 \\\\ 0 &#038; 1 &#038; -1 \\end{array}\\right)<br \/>\n\\left(\\begin{array}{} x_1 \\\\ x_2 \\\\ x_3 \\end{array}\\right)<br \/>\n\\]<br \/>\n\uacfc \uac19\uc774 \ud589\ub82c\uc744 \uc67c\ucabd\uc5d0 \uacf1\ud568\uc73c\ub85c\uc368 (2.1)\uacfc \uac19\uc740 \uc120\ud615\ubcc0\ud658\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \ub0b4\uc6a9\uc744 \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2 marginbottom2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.<\/span><br \/>\n\uc120\ud615\ubcc0\ud658 \\(T\\in\\Hom (K^n ,\\, K^m )\\)\uacfc \ud589\ub82c \\(M(T)\\in\\Mat_{m\\times n} (K)\\)\ub97c \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 \ub3d9\ud615\uc0ac\uc0c1\uc774\uba70, \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(\\Hom (K^n ,\\, K^m )\\)\uacfc \\(\\Mat_{m\\times n} (K)\\)\ub294 \ub3d9\ud615\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc815\ub9ac 2\ub294 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \ud589\ub82c \\(A\\in\\Mat_{m\\times n} (K)\\)\uc5d0 \ub300\uc751\ub418\ub294 \uc120\ud615\ubcc0\ud658\uc744 \\(T_A\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[M(T_A) = A\\tag{2.3}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ed\uc73c\ub85c \uc784\uc758\uc758 \\(T\\in\\Hom (K^n ,\\, K^m )\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[T_{M(T)} = T\\tag{2.4}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc9c0\uae08\ubd80\ud130\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc740 \\(m\\times n\\) \ud589\ub82c\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc5ec\uaca8\ub3c4 \uc88b\ub2e4. (\\(m\\)\uacfc \\(n\\)\uc758 \uc21c\uc11c\ub97c \uc8fc\uc758\ud558\uc790.)<\/p>\n<p>\uc815\ub9ac 2\ub85c\ubd80\ud130 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"corollary margintop2 marginbottom2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 3.<\/span><br \/>\n\\(\\Hom(K^n ,\\, K^m )\\)\uc758 \ucc28\uc6d0\uc740 \\(mn\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uc740 \ud589\ub82c\uc758 \uacf1\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \uc774 \ub0b4\uc6a9\uc744 \ub354 \uc790\uc138\ud788 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[A\\in\\Mat_{m\\times n}(K) ,\\,\\, B\\in\\Mat_{n\\times p} (K)\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(T_{AB}\\)\uc640 \\(T_A \\circ T_B\\)\ub294 \ubaa8\ub450 \\(\\Hom(K^p ,\\, K^m )\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[T_{AB}(\\mathbf{e}_j) = (AB)\\mathbf{e}_j = (AB)^j\\tag{2.5}\\]<br \/>\n\uc774\uace0<br \/>\n\\[\\begin{align}<br \/>\n(T_A \\circ T_B )(\\mathbf{e}_j) &#038;= T_A (T_B (\\mathbf{e}_j )) = T_A(B\\mathbf{e}_j) \\\\[5pt]<br \/>\n&#038;= T_A(B^j ) = A\\cdot B^j\\tag{2.6}\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 (2.5)\uc640 (2.6)\uc740 \ubaa8\ub450 \\(AB\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\" style=\"padding-bottom: 0;\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.<\/span><br \/>\n\\(A\\in\\Mat_{m\\times n}(K) ,\\) \\(B\\in\\Mat_{n\\times p} (K)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\[T_{AB} = T_A \\circ T_B .\\]\n<\/p><\/div>\n<p>\uc989 \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uc740 \ud589\ub82c\uc758 \uacf1\uacfc \uac19\ub2e4. \uadf8\ub7f0\ub370 \ud568\uc218\uc758 \ud569\uc131\uc758 \uacb0\ud569\ubc95\uce59\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"corollary margintop2 marginbottom2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 5.<\/span><br \/>\n\ud589\ub82c\uc758 \uacf1\uc758 \uacb0\ud569\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774 \ub530\ub984\uc815\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"corollary margintop2 marginbottom2\" style=\"padding-bottom: 0;\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 6.<\/span><br \/>\n\\(T : K^p \\rightarrow K^n \\)\uacfc \\(T &#8216; : K^n \\rightarrow K^m\\)\uc774 \uc120\ud615\ubcc0\ud658\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\[M(T &#8216; \\circ T ) = M(T &#8216; ) M(T) .\\]\n<\/p><\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\\(V = \\mathbb{R}^2\\)\uc774\uace0 \\(T_\\theta\\)\uac00 \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \uc2dc\uacc4 \ubc18\ub300\ubc29\ud5a5\uc73c\ub85c \\(\\theta\\)\ub9cc\ud07c \ud68c\uc804\ud558\ub294 \ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \ub450 \ud45c\uc900\uae30\uc9c0\uc6d0\uc18c\ub97c \\(\\theta\\)\ub9cc\ud07c \ud68c\uc804\ud55c \uacb0\uacfc\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nT_\\theta(\\mathbf{e}_1) &#038;= (\\cos\\theta ,\\, \\sin\\theta ), \\\\[5pt]<br \/>\nT_\\theta(\\mathbf{e}_2) &#038;= ( -\\sin\\theta ,\\, \\cos\\theta ).<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(T_\\theta\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[M(T_\\theta ) = \\left(\\begin{array}{cr}<br \/>\n\\cos\\theta &#038; -\\sin\\theta \\\\<br \/>\n\\sin\\theta &#038; \\cos\\theta<br \/>\n\\end{array}\\right).\\]<br \/>\n\uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \\(\\beta\\)\ub9cc\ud07c \ud68c\uc804\ud55c \ud6c4 \\(\\alpha\\)\ub9cc\ud07c \ub354 \ud68c\uc804\ud558\ub294 \uac83\uc740 \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \\(\\alpha+\\beta\\)\ub9cc\ud07c \ud68c\uc804\ud558\ub294 \uac83\uacfc \uac19\uc73c\ubbc0\ub85c<br \/>\n\\[T_{\\alpha+\\beta} = T_\\alpha \\circ T_\\beta\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[M(T_{\\alpha+\\beta} ) = M(T_\\alpha)M(T_\\beta)\\]<br \/>\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uc744 \ud589\ub82c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[<br \/>\n\\left(\\begin{array}{cr}<br \/>\n\\cos(\\alpha+\\beta) &#038; -\\sin(\\alpha+\\beta) \\\\<br \/>\n\\sin(\\alpha+\\beta) &#038; \\cos(\\alpha+\\beta)<br \/>\n\\end{array}\\right)<br \/>\n=<br \/>\n\\left(\\begin{array}{cr}<br \/>\n\\cos\\alpha &#038; -\\sin\\alpha \\\\<br \/>\n\\sin\\alpha &#038; \\cos\\alpha<br \/>\n\\end{array}\\right)<br \/>\n\\left(\\begin{array}{cr}<br \/>\n\\cos\\beta &#038; -\\sin\\beta \\\\<br \/>\n\\sin\\beta &#038; \\cos\\beta<br \/>\n\\end{array}\\right)\\]<br \/>\n\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc758 \ud589\ub82c\uacf1\uc744 \ud480\uc5b4\uc11c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\cos(\\alpha+\\beta) &#038;= \\cos\\alpha \\cos\\beta &#8211; \\sin\\alpha \\sin\\beta , \\\\[5pt]<br \/>\n\\sin(\\alpha+\\beta) &#038;= \\sin\\alpha \\cos\\beta + \\cos\\alpha \\sin\\beta .<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\\(\\Hom(V ,\\, V &#8216; )\\)\uc758 \ud45c\ud604<\/h2>\n<p>\uc9c0\uae08\ubd80\ud130 \ub2e4\ub8e8\ub294 \ubca1\ud130\uacf5\uac04\uc740 \ubcc4\ub2e4\ub978 \uc5b8\uae09\uc774 \uc5c6\ub294 \ud55c \uc720\ud55c\ucc28\uc6d0\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud558\uc790.<\/p>\n<p>\\(V\\)\uac00 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(B = \\left\\{ v_1 ,\\, \\cdots ,\\, v_n \\right\\}\\)\uc774 \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc88c\ud45c\ud568\uc218 \\(\\gamma_B : V \\rightarrow K^n\\)\uc740<br \/>\n\\[\\gamma_B \\left( \\sum_{j=1}^n \\lambda_j v_j \\right) =(\\lambda_1 ,\\, \\cdots ,\\, \\lambda_n )\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ub41c \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(V &#8216; \\)\uc774 \\(m\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc77c \ub54c \uc88c\ud45c\ud568\uc218 \\(\\gamma_{B &#8216; } : V &#8216; \\rightarrow K^m\\)\uc740 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<\/p>\n<p>\uc774\ub54c \\(V\\)\ub85c\ubd80\ud130 \\(V &#8216; \\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658 \\(T : V \\rightarrow V &#8216; \\)\uc740 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658<br \/>\n\\[\\gamma_{B &#8216; } \\circ T \\circ \\gamma_B ^{-1} : K^n \\rightarrow K^m \\tag{3.1}\\]<br \/>\n\uc5d0 \ub300\uc751\ub41c\ub2e4. \uc774\ub7ec\ud55c \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uc744 <span class=\"defined\">\uae30\uc800 \\(B,\\) \\(B &#8216; \\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(M_{B,B &#8216; } (T)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc120\ud615\ubcc0\ud658 \\(T : V \\rightarrow V &#8216; \\)\uc744 (3.1)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\uba70 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(\\Hom(V ,\\, V &#8216; )\\)\uacfc \\(\\Hom(K^n ,\\, K^m )\\)\uc740 \ub3d9\ud615\uc774\ub2e4. \ub354\uc6b1\uc774 \\(\\Hom(K^n ,\\, K^m )\\)\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \\(\\Hom(V ,\\, V &#8216; )\\)\uc758 \uc131\uc9c8\uc744 \ub04c\uc5b4\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\\(V\\)\uc640 \\(V &#8216; \\)\uc774 \uac01\uac01 \\(n\\)\ucc28\uc6d0, \\(m\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uba74 \\(\\Hom(V,\\,V &#8216; )\\)\uc740 \\(mn\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.<\/li>\n<li>\\(T : V \\rightarrow V &#8216; \\)\uacfc \\(T &#8216; : V &#8216; \\rightarrow V &#8221; \\)\uc774 \uc120\ud615\ubcc0\ud658\uc774\uba74 \\(T &#8216; \\circ T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.\\[M_{B,B &#8216; } (T &#8216; \\circ T ) = M_{B &#8216;, B &#8216; &#8216;} (T &#8216; ) M_{B,B &#8216;} (T ).\\]<\/li>\n<\/ul>\n<p>\\(V\\)\uac00 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc778 \uacbd\uc6b0 \\(\\Hom(V,\\,V)\\)\ub294 \\(\\Hom(K^n ,\\, K^n)\\)\uacfc \ub3d9\ud615\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\Hom(K^n ,\\, K^n)\\)\uc774 K-\ub300\uc218\uc774\ubbc0\ub85c \\(\\Hom(V,\\,V)\\)\ub3c4 K-\ub300\uc218\uc774\ub2e4. \\(V\\)\uc758 \uae30\uc800 \\(B\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \\(M_{B,B}(T)\\)\ub97c \uac04\ub2e8\ud788 \\(M_B(T)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \\(T\\in\\Hom(V,\\,V)\\)\ub97c \\(\\gamma_B \\circ T \\circ \\gamma_B^{-1} \\in \\Hom(K^n ,\\, K^n)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 \ubca1\ud130\ub3d9\ud615\uc0ac\uc0c1\uc774\uba70 \ud568\uc218\uc758 \ud569\uc131\uc744 \ubcf4\uc874\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\Hom(V,\\,V)\\)\uc640 \\(\\Hom(K^n,\\,K^n)\\)\uc740 K-\ub300\uc218 \ub3d9\ud615\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\\(V\\)\uac00 \\(\\sin x\\)\uc640 \\(\\cos x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \uc5f0\uc18d\ud568\uc218\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(\\sin x\\)\uc640 \\(\\cos x\\)\uc740 \uadf8 \uc790\uccb4\ub85c\uc11c \\(V\\)\uc758 \uae30\uc800\uac00 \ub41c\ub2e4. \\(D\\)\uac00 \ubbf8\ubd84\uc5f0\uc0b0\uc790\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\begin{align}<br \/>\nD(\\sin x) &#038;= 0 \\cdot \\sin x + 1 \\cdot \\cos x ,\\\\[5pt]<br \/>\nD(\\cos x) &#038;= -1 \\cdot \\sin x + 0 \\cdot \\cos x<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c \\(D\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[A = \\left(\\begin{array}{cr} 0 &#038; -1 \\\\ 1 &#038; 0 \\end{array}\\right).\\]<br \/>\n\ub9cc\uc57d \\(f(x) = 2\\sin x + 5\\cos x\\)\uc774\uba74 \\(f\\)\ub294 \uc21c\uc11c\uc30d \\((2,\\,5)\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ubca1\ud130\uc774\uace0<br \/>\n\\[\\left(\\begin{array}{cr} 0 &#038; -1 \\\\ 1 &#038; 0 \\end{array}\\right)<br \/>\n\\left(\\begin{array}{} 2\\\\5 \\end{array}\\right) =<br \/>\n\\left(\\begin{array}{r} -5\\\\2 \\end{array}\\right)\\]<br \/>\n\uc774\ubbc0\ub85c \\(Df\\)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[D(f(x)) = f &#8216; (x) = -5 \\sin x + 2 \\cos x\\]<br \/>\n\ud55c\ud3b8 \\(D^2 = D \\circ D\\)\uc774\ubbc0\ub85c \\(D^2\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[A^2 = \\left(\\begin{array}{rr} -1 &#038; 0 \\\\ 0 &#038; -1 \\end{array}\\right).\\]<br \/>\n\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(D^4\\)\uc758 \ud45c\ud604\ud589\ub82c\uc744 \uad6c\ud558\uba74 \\(A^4 = A^2 A^2 = I_2\\)\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<h2 class=\"margintop2\">\uc30d\ub300\uacf5\uac04<\/h2>\n<p>\\(V\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\ub294 \uccb4\uc774\uc9c0\ub9cc 1\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uae30\ub3c4 \ud558\ub2e4. \uc774\ub54c \\(V\\)\ub85c\ubd80\ud130 \\(K\\)\ub85c\uc758 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ubaa8\uc784\uc744 \\(V^*\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \\(V\\)\uc758 <span class=\"defined\">\uc30d\ub300\uacf5\uac04<\/span>(dual space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(V^* = \\Hom(V,\\,K)\\)\uc774\ub2e4. \uadf8\ub9ac\uace0 \\(V^*\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\uc120\ud615\ubc94\ud568\uc218<\/span>(linear functional)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(V\\)\uac00 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uba74 \\(V^*\\) \ub610\ud55c \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\ub2e4. (\ucc38\uace0:<br \/>\n<a href=\"\/blog\/articles\/linear-algebra-dual-basis\/\" target=\"_blank\" rel=\"noopener noreferrer\">\uc30d\ub300\uacf5\uac04\uc758 \uae30\uc800<\/a>)<\/p>\n<p>\uc30d\ub300\uacf5\uac04\uc758 \uc815\uc758\ub9cc \ubcf4\uc558\uc744 \ub550 \uadf8\ub2e4\uc9c0 \uc4f8\ubaa8\uc788\uc5b4 \ubcf4\uc774\uc9c0 \uc54a\ub294\ub2e4. \ud558\uc9c0\ub9cc \uc30d\ub300\uacf5\uac04\uc740 \ud589\ub82c\uc758 \ud589\uacf5\uac04\uc758 \uc131\uc9c8\uacfc \uc5f4\uacf5\uac04\uc758 \uc131\uc9c8\uc744 \uc774\uc5b4\uc8fc\ub294 \uc5ed\ud560\uc744 \ud55c\ub2e4. \uc9c0\uae08\ubd80\ud130 \uadf8 \uacfc\uc815\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\\(V,\\) \\(W\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \uc120\ud615\ubcc0\ud658 \\(T : V \\rightarrow W\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(V\\)\uc640 \\(W\\)\uc758 \uc30d\ub300\uacf5\uac04\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub294 \uac83\ucc98\ub7fc \\(T\\)\uc5d0 \ub300\uc751\ub418\ub294 \uc120\ud615\ubcc0\ud658 \\(T^*\\)\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \\(T\\)\ub294 \ubca1\ud130\ub97c \ubca1\ud130\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ubcc0\ud658\uc774\uc9c0\ub9cc \\(T^*\\)\ub294 \ubc94\ud568\uc218\ub97c \ubc94\ud568\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ubcc0\ud658\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n<p>\\(f\\in W^*\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\ub97c \\(f\\circ T\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218<br \/>\n\\[T^* : W^* \\rightarrow V^* ,\\,\\, f\\mapsto f\\circ T\\]<br \/>\n\ub97c \\(T\\)\uc758 <span class=\"defined\">\uc804\uce58\ud568\uc218<\/span>(transpose map)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc804\uce58\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\\(K\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0 \ub300\ud558\uc5ec \\((1_V)^* = 1_{V^*}\\)\uc774\ub2e4. \uc989 \uc804\uce58\ud568\uc218\ub294 \ud56d\ub4f1\ud568\uc218\ub97c \ud56d\ub4f1\ud568\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4.<\/li>\n<li>\\(T_1\\)\uacfc \\(T_2\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04 \uc704\uc5d0\uc11c\uc758 \uc120\ud615\ubcc0\ud658\uc774\uace0 \\(T_1 \\circ T_2\\)\uac00 \uc815\uc758\ub418\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[(T_1 \\circ T_2 )^* = T_2 ^* \\circ T_1 ^* .\\]<br \/>\n\uc774\uc640 \uac19\uc740 \uc131\uc9c8\uc744 <span class=\"defined\">\ubc18\ubcc0\uc131\uc9c8<\/span>(contravariance)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<p>\ub354\uc6b1\uc774 \\(T\\)\uc758 \ub300\uc751 \uc131\uc9c8\uacfc \\(T^*\\)\uc758 \ub300\uc751 \uc131\uc9c8\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\ub97c \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\\(T\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74 \\(T^*\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\\(T\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\uba74 \\(T^*\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\\(T\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc774\uba74 \\(T^*\\)\ub3c4 \uc77c\ub300\uc77c \ub300\uc751\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub7ec\ud55c \uc131\uc9c8\uc758 \ub530\ub984\uc815\ub9ac\ub85c\uc11c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.<\/span><br \/>\n\\(V\\)\uc640 \\(W\\)\uac00 \\(K\\) \uc704\uc5d0\uc11c\uc758 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T : V \\rightarrow W\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\operatorname{rk}(T) = \\operatorname{rk}(T^* ).\\]<br \/>\n\uc989 \\(T\\)\uc758 \uc0c1(image)\uc758 \ucc28\uc6d0\uacfc \\(T^*\\)\uc758 \uc0c1\uc758 \ucc28\uc6d0\uc774 \uac19\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(T\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc778 \uacbd\uc6b0\ubd80\ud130 \uc0b4\ud3b4\ubcf4\uc790. \uadf8\ub7ec\uba74 \\(\\operatorname{Im} (T) = W\\)\uc774\uace0 \\(\\operatorname{rk}(T) = \\dim(W)\\)\uc774\ub2e4. \\(T^* : W^* \\rightarrow V^*\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \\(T^*\\)\uc758 \ud575\uc758 \ucc28\uc6d0\uc740 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c Rank-Nullity \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\operatorname{rk}(T^*) = \\dim(W^*)\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\dim(W^*) = \\dim(W)\\)\uc774\ubbc0\ub85c \\[\\operatorname{rk}(T) = \\dim(W) = \\dim(W^*) = \\operatorname{rk}(T^*)\\]\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(T\\)\uac00 \uc77c\ubc18\uc801\uc778 \uc120\ud615\ubcc0\ud658\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(T\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uac00 \uc544\ub2d0 \uc218 \uc788\uae30 \ub54c\ubb38\uc5d0, \\(T\\)\uc640 \uac19\uc740 \uc5ed\ud560\uc744 \ud558\ub294 \uc704\ub85c\uc758 \ud568\uc218 \\(T_1\\)\uc744 \uad6c\uc131\ud574\uc57c \ud55c\ub2e4. \\(\\operatorname{Im}(T) = W_1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud3ec\ud568\uc0ac\uc0c1 \\(i : W_1 \\rightarrow W\\)\uc5d0 \ub300\ud558\uc5ec \\(i\\circ T_1\\)\uc740 \\(V\\)\ub85c\ubd80\ud130 \\(W\\)\ub85c\uc758 \ud568\uc218\uc774\uace0<br \/>\n\\[T = i\\circ T_1\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(T_1\\)\uc740 \\(T\\)\uc640 \ub3d9\uc77c\ud558\uac8c \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\uace0 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba70 \\(\\operatorname{rk}(T) = \\dim(W_1)= \\operatorname{rk}(T_1)\\)\uc774\ub2e4. \\(T,\\) \\(T_1,\\) \\(i\\)\uc758 \uc804\uce58\ud568\uc218\ub97c \uc0dd\uac01\ud558\uba74 \\(T^*\\)\uc640 \\(T_1^* \\circ i^*\\)\ub294 \ubaa8\ub450 \\(W^*\\)\ub85c\ubd80\ud130 \\(V^*\\)\ub85c\uc758 \ud568\uc218\uc774\uace0<br \/>\n\\[T_1^* \\circ i^* = T^*\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(i^*\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\ubbc0\ub85c \\(T^*\\)\uc640 \\(T_1^*\\)\ub294 \uac19\uc740 \uc0c1\uc744 \uac00\uc9c0\uba70, \uac19\uc740 \uacc4\uc218\ub97c \uac00\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\operatorname{rk}(T) = \\operatorname{rk}(T_1) \\,\\text{ and }\\, \\operatorname{rk}(T^*) = \\operatorname{rk}(T_1 ^*)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(T_1\\)\uc774 \uc704\ub85c\uc758 \ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[\\operatorname{rk}(T_1) = \\operatorname{rk}(T_1 ^*)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\operatorname{rk}(T) = \\operatorname{rk}(T^* )\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(B_1 = \\left\\{ v_1 ,\\, \\cdots ,\\, v_n \\right\\}\\)\uc774 \\(V\\)\uc758 \uae30\uc800\uc774\uace0 \\(B_2 = \\left\\{ w_1 ,\\, \\cdots ,\\, w_m \\right\\}\\)\uc774 \\(V_2\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uc774 \ub450 \uae30\uc800\uc758 \uc30d\ub300\uae30\uc800\ub97c \\(B_1 ^* ,\\) \\(B_2 ^*\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uc774\ub54c \uc804\uce58\ud568\uc218 \\(T^*\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \uc120\ud615\ubcc0\ud658 \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.<\/span><br \/>\n\uae30\uc800 \\(B_1 ,\\) \\(B_2\\)\uc5d0 \ub300\ud55c \uc120\ud615\ubcc0\ud658 \\(T:V_1 \\rightarrow V_2\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \\(M(T)\\)\uc774\uba74, \uae30\uc800 \\(B_2 ^*,\\) \\(B_1 ^*\\)\uc5d0 \ub300\ud55c \uc804\uce58\ud568\uc218 \\(T^* : V_2^* \\rightarrow V_1^*\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[M(T^*) = \\, ^t \\! M(T).\\]<br \/>\n\uc989 \uc120\ud615\ubcc0\ud658 \\(T\\)\uc758 \uc804\uce58\ud568\uc218\uc758 \ud45c\ud604\ud589\ub82c\uc740 \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc758 \uc804\uce58\ud589\ub82c\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uae30\uc800 \\(B_2 ^*,\\) \\(B_1 ^*\\)\uc5d0 \ub300\ud55c \\(T^*\\)\uc758 \ud45c\ud604\ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\ub294 \\(v_1^*,\\) \\(\\cdots,\\) \\(v_n^*\\)\uc5d0 \ub300\ud55c \\(T^*(w_j ^*)\\)\uc758 \uc88c\ud45c\ubca1\ud130\uc640 \uac19\ub2e4. \uc774\uac83\uc740 \ub2e4\uc74c \uc138 \uac00\uc9c0 \uc0ac\uc2e4\uc744 \uc5fc\ub450\uc5d0 \ub450\uba74 \uc27d\uac8c \uacc4\uc0b0\ub41c\ub2e4.<\/p>\n<div>\n<ol class=\"bracket\">\n<li>\uc120\ud615\ubcc0\ud658 \\(f:V\\rightarrow K\\)\ub294 \uc30d\ub300\uae30\uc800\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.\\[f = \\sum_{j=1}^n f(v_j)v_j^*\\]<\/li>\n<li>\uae30\uc800 \\(B_1,\\) \\(B_2\\)\uc5d0 \ub300\ud558\uc5ec \\(M(T) = (a_{ij})\\)\uc774\uba74\\[T(v_j) = \\sum_{i=1}^m a_{ij}w_i\\]\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(M(T)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\ub294 \\(T(v_j)\\)\ub97c \\(w_1,\\) \\(\\cdots,\\) \\(w_m\\)\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ud558\uae30 \uc704\ud55c \uacc4\uc218\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<li>\uc30d\ub300\uae30\uc800\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec\\[w_j^* \\left( \\sum_{i=1}^m \\lambda_i w_i \\right) = \\lambda_j\\]\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(w_j^*\\)\ub294 \\(w_j\\)\ub97c \uc81c\uc678\ud55c \ub2e4\ub978 \uae30\uc800\uc6d0\uc18c\ub97c \\(0\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc774\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \\(T^* (w_j^*)\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nT^*(w_j^*) &#038;= w_j^* \\circ T = \\sum _{i=1}^n [ w_j^* \\circ T](v_i)v_i^* \\\\[5pt]<br \/>\n&#038;=\\sum_{i=1}^n w_j^* (T(v_i ))v_i^* = \\sum _{i=1}^n w_j^* \\left( \\sum_{k=1}^m a_{ki} w_k \\right) v_i^*\\\\[5pt]<br \/>\n&#038;= \\sum_{i=1}^n a_{ji} v_i^*.<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub85c\uc368 \\(M(T^*)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uc640 \\(M(T)\\)\uc758 \\(j\\)\uc9f8 \ud589\ubca1\ud130\uac00 \uc77c\uce58\ud558\ubbc0\ub85c \\(M(T^*) = \\,^t \\! M(T)\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\ub85c\uc368 \ud589\ub82c\uc758 \ud589\ubca1\ud130\uac00 \uac00\uc9c0\ub294 \uc131\uc9c8\uacfc \uc5f4\ubca1\ud130\uac00 \uac00\uc9c0\ub294 \uc131\uc9c8\uc744 \uc11c\ub85c \uad50\ud658\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 7.<\/span><br \/>\n\ud589\ub82c\uc758 \ud589\uacf5\uac04\uc758 \ucc28\uc6d0\uacfc \uc5f4\uacf5\uac04\uc758 \ucc28\uc6d0\uc740 \uac19\ub2e4.\n<\/p>\n<\/div>\n<p>\ub354\uc6b1\uc774 \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 8.<\/span><br \/>\n\\(A\\in M_n (K)\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc740 \ubaa8\ub450 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"bracket marginbottom0\">\n<li>\uc784\uc758\uc758 \\(\\mathbf{y}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{y}\\)\ub294 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \ud574\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc744 \uc0dd\uc131\ud55c\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc744 \uc0dd\uc131\ud55c\ub2e4.<\/li>\n<li>\ub3d9\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{0}\\)\uc740 \uc790\uba85\ud55c \ud574 \\(\\mathbf{x} = \\mathbf{0}\\)\ub9cc\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\mathbf{y}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{y}\\)\ub294 \ub531 \ud558\ub098\uc758 \ud574\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc758 \uae30\uc800\ub97c \uc774\ub8ec\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc758 \uae30\uc800\ub97c \uc774\ub8ec\ub2e4.<\/li>\n<li>\\(A\\)\ub294 \uac00\uc5ed\uc774\ub2e4. \uc989 \\(A\\in\\operatorname{GL}_n (K)\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h2 class=\"margintop2\">\uc774\uc911\uc30d\ub300\uacf5\uac04<\/h2>\n<p>\\(V\\)\uac00 \ubca1\ud130\uacf5\uac04\uc77c \ub54c \\(V\\)\uc758 \uc30d\ub300\uacf5\uac04\uc758 \uc30d\ub300\uacf5\uac04\uc744 \\(V\\)\uc758 <span class=\"defined\">\uc774\uc911\uc30d\ub300\uacf5\uac04<\/span>(double dual space)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(V^{**}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\\(V\\)\uc640 \\(V^*\\)\uac00 \ub3d9\ud615\uc784\uc744 \ubcf4\uc77c \ub54c \\(V\\)\uc758 \uae30\uc800\ub97c \uc774\uc6a9\ud558\uc5ec \uc30d\ub300\uae30\uc800\ub97c \uad6c\uc131\ud55c \ub4a4, \uadf8\uac83\uc774 \\(V^*\\)\uc758 \uae30\uc800\uac00 \ub428\uc744 \ubcf4\uc600\ub2e4. \uadf8\ub7ec\ub098 \\(V\\)\uc640 \\(V^{**}\\)\uc774 \ub3d9\ud615\uc784\uc744 \ubcf4\uc77c \ub550 \\(V\\)\uc758 \uae30\uc800\uc640 \uc0c1\uad00 \uc5c6\uc774 \ub450 \uacf5\uac04 \uc0ac\uc774\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\\(v\\in V\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc784\uc758\uc758 \\(f\\in V^*\\)\ub97c \\(f(v)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218 \\(e_v : V^* \\rightarrow K\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc989 \\(e_v (f) = f(v)\\)\uc774\ub2e4. \uc774 \ud568\uc218\ub294 \\(V^*\\)\ub85c\ubd80\ud130 \\(K\\)\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774 \ub41c\ub2e4. \uc989 \\(e_v \\in V^{**}\\)\uc774\ub2e4. \ub9cc\uc57d<br \/>\n\\[v \\mapsto e_v\\]<br \/>\n\ub85c\uc11c \\(V\\)\uc758 \uc6d0\uc18c\ub97c \\(V^{**}\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\uba74, \uc774\ub7ec\ud55c \ub300\uc751\uc740 \ub450 \uacf5\uac04 \uc0ac\uc774\uc758 \uc120\ud615\ubcc0\ud658\uc774\uba74\uc11c \uc77c\ub300\uc77c \ub300\uc751\uc774 \ub41c\ub2e4. \uc774\ub807\uac8c \ub450 \uacf5\uac04 \uc0ac\uc774\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774 \uae30\uc800\uc640 \ubb34\uad00\ud558\uac8c \uc815\ud574\uc9c0\ub294\ub370, \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \u2018\\(V\\)\uc640 \\(V^{**}\\)\ub294 <span class=\"defined\">\ud45c\uc900\uc801\uc73c\ub85c \ub3d9\ud615\uc774\ub2e4<\/span>(canonically isomorphic)\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<h2 class=\"margintop2\">\uae30\uc800\uc758 \ubcc0\ud658<\/h2>\n<p>\ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800\uac00 \uc5b4\ub290\uac83\uc774 \uc8fc\uc5b4\uc84c\ub290\ub0d0\uc5d0 \ub530\ub77c\uc11c \uadf8 \ubca1\ud130\uacf5\uac04 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uc774 \ub2ec\ub77c\uc9c4\ub2e4. \ubca1\ud130\uacf5\uac04\uc758 \uae30\uc800\uac00 \ub2e4\ub978 \uac83\uc73c\ub85c \ubc14\ub00c\uc5c8\uc744 \ub54c \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uc774 \uc5b4\ub5bb\uac8c \ub2ec\ub77c\uc9c0\ub294\uc9c0 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\\(B = \\left\\{v_1 ,\\, \\cdots ,\\, v_n\\right\\}\\)\uacfc \\(B &#8216; = \\left\\{ w_1 ,\\, \\cdots ,\\, w_n \\right\\}\\)\uc774 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc88c\ud45c\ud568\uc218 \\(\\gamma_B\\)\uc640 \\(\\gamma_{B &#8216; }\\)\uc740 \ubaa8\ub450 \\(V\\)\ub85c\ubd80\ud130 \\(K^n\\)\uc73c\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \uc774\ub54c \\(P = \\gamma_B \\circ \\gamma_{B &#8216;} ^{-1}\\)\ub77c\uace0 \ud558\uba74 \\(P\\)\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^n\\)\uc73c\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \ub354\uc6b1\uc774 \\(P\\)\ub294 \uae30\uc800 \\(B &#8216; \\)\uc5d0 \uc758\ud558\uc5ec \ud45c\ud604\ub41c \\(V\\)\uc758 \ubca1\ud130\ub97c \uae30\uc800 \\(B\\)\uc5d0 \uc758\ud558\uc5ec \ud45c\ud604\ub41c \\(V\\)\uc758 \ub3d9\uc77c\ud55c \ubca1\ud130\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(P\\)\ub97c \\(B &#8216; \\)\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uae30\uc800\ubcc0\ud658\ud589\ub82c\uacfc \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc774 \ub098\ud0c0\ub0b4\ub294 \ud568\uc218\ub97c \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec, \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218\ub97c \\(T_P\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \uadf8 \ud589\ub82c\uc744 \\(P\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uac01 \\(j = 1,\\,\\cdots,\\,n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[P\\gamma_{B&#8217;} (w_j) = \\gamma_B (w_j)\\]<br \/>\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc740 \\(P\\mathbf{e}_j = P^j\\)\uc774\uace0 \uc6b0\ubcc0\uc740 \\(w_j\\)\uc758 \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \uc88c\ud45c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uae30\uc800\ubcc0\ud658\ud589\ub82c \\(P\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box\">\n<p>\uae30\uc800\ubcc0\ud658\ud589\ub82c \\(P\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc740 \\(w_j\\)\uc758 \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \uc88c\ud45c\ubca1\ud130\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\\(V\\)\uac00 \ucc28\uc218\uac00 2 \uc774\ud558\uc778 \uc2e4\uacc4\uc218 \ub2e4\ud56d\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uc758 \ub450 \uae30\uc800<br \/>\n\\[B = \\left\\{ 1,\\,x,\\,x^2 \\right\\} ,\\,\\, B &#8216; = \\left\\{ 1+x ,\\, 1-x ,\\, 1+x^2 \\right\\}\\]<br \/>\n\uc744 \uc0dd\uac01\ud558\uc790. \uc774\ub54c \\(B &#8216; \\)\uc73c\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[P = \\left(\\begin{array}{crc}<br \/>\n1 &#038; 1 &#038; 1 \\\\ 1 &#038; -1 &#038; 0 \\\\ 0 &#038; 0 &#038; 1<br \/>\n\\end{array}\\right).\\]\n<\/p>\n<\/div>\n<div class=\"theorem margintop2\" style=\"padding-bottom: 0;\">\n<p><span class=\"theorem\">\uc815\ub9ac 9.<\/span><br \/>\n\\(V\\)\uac00 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(T:V\\rightarrow V\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(B\\)\uc640 \\(B &#8216; \\)\uc774 \\(V\\)\uc758 \uae30\uc800\uc774\uba70 \\(P\\)\uac00 \\(B &#8216; \\)\uc73c\ub85c\ubd80\ud130 \\(B\\)\ub85c\uc758 \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \\(M\\)\uc774\uace0, \uae30\uc800 \\(B &#8216; \\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774 \\(N\\)\uc774\uba74, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[N = P^{-1} MP.\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\ud654\ub048\ud558\uac8c \ub2e4\uc774\uc5b4\uadf8\ub7a8 \ud55c \uc7a5\uc73c\ub85c \uc99d\uba85\uc744 \ub9c8\uce5c\ub2e4.<br \/>\n<img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis.png\" alt=\"\" width=\"301\" height=\"140\" class=\"aligncenter size-full wp-image-5568\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis.png 1202w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis-300x140.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis-1024x479.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis-768x359.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis-1170x547.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2020\/10\/change_of_basis-585x274.png 585w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><br \/>\n(\uc368 \ub193\uace0 \ubcf4\ub2c8 \\(K^n\\)\uacfc \\(K^n\\) \uc0ac\uc774\uc5d0 \\(M\\)\uc774 \ube60\uc84c\ub124.)<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example margintop2\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\ubcf4\uae30 4\uc758 \ub0b4\uc6a9\uc744 \uc774\uc5b4\uc11c \uc0b4\ud3b4\ubcf4\uc790. \uae30\uc800\ub97c \\(B\\)\uc5d0\uc11c \\(B &#8216; \\)\uc73c\ub85c \ubc14\uafb8\uc5c8\uc744 \ub54c \ubbf8\ubd84\uc5f0\uc0b0\uc790\uc758 \ud45c\ud604\ud589\ub82c\uc744 \uad6c\ud574 \ubcf4\uc790. \uba3c\uc800 \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc758 \uc5ed\ud589\ub82c\uc744 \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[P^{-1} = \\left(\\begin{array}{crr}<br \/>\n1\/2 &#038; 1\/2 &#038; -1\/2 \\\\<br \/>\n1\/2 &#038; -1\/2 &#038; -1\/2 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{array}\\right).\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \uae30\uc800 \\(B &#8216; \\)\uc5d0 \ub300\ud55c \ubbf8\ubd84\uc5f0\uc0b0\uc790 \\(D\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[P^{-1}DP =<br \/>\n\\left(\\begin{array}{crr}<br \/>\n1\/2 &#038; 1\/2 &#038; -1\/2 \\\\<br \/>\n1\/2 &#038; -1\/2 &#038; -1\/2 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{array}\\right)<br \/>\n\\left(\\begin{array}{ccc}<br \/>\n0 &#038; 1 &#038; 0 \\\\ 0 &#038; 0 &#038; 2 \\\\ 0 &#038; 0 &#038; 0<br \/>\n\\end{array}\\right)<br \/>\n\\left(\\begin{array}{crc}<br \/>\n1 &#038; 1 &#038; 1\\\\ 1 &#038; -1 &#038; 0 \\\\ 0 &#038; 0 &#038; 1<br \/>\n\\end{array}\\right).<br \/>\n\\]<br \/>\n\uc774\uac83\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[P^{-1}DP =<br \/>\n\\left(\\begin{array}{crr}<br \/>\n1\/2 &#038; -1\/2 &#038; 1 \\\\ 1\/2 &#038; -1\/2 &#038; -1 \\\\ 0 &#038; 0 &#038; 0<br \/>\n\\end{array}\\right).<br \/>\n\\]\n<\/p>\n<\/div>\n<p>\ub450 \ud589\ub82c \\(A,\\,B \\in M_n (K)\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uac00\uc5ed\ud589\ub82c \\(P\\in\\operatorname{GL}_n (K)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B = P^{-1}AP\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74, \ub450 \ud589\ub82c\uc744 \uc11c\ub85c <span class=\"defined\">\ub2ee\uc740\ud589\ub82c<\/span>(similar)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(A \\sim B\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ud589\ub82c\uc758 \ub2ee\uc74c \uad00\uacc4\ub294 \ub3d9\uce58\uad00\uacc4\uc774\ub2e4. \uc989 \ud589\ub82c\uc758 \ub2ee\uc74c \uad00\uacc4\ub294 \ubc18\uc0ac\uc801\uc774\uace0 \ubc18\ub300\uce6d\uc801\uc774\uba70 \ucd94\uc774\uc801\uc774\ub2e4. \ub354\uc6b1\uc774 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 10.<\/span><br \/>\n\ub450 \ud589\ub82c \\(M,\\,M &#8216; \\in M_n (K)\\)\uac00 \uc11c\ub85c \ub2ee\uc74c\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc640 \uc120\ud615\ubcc0\ud658 \\(T\\), \uadf8\ub9ac\uace0 \ub450 \uae30\uc800 \\(B,\\) \\(B &#8216; \\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(T\\)\uc758 \\(B ,\\) \\(B &#8216; \\)\uc5d0 \ub300\ud55c \ud45c\ud604\ud589\ub82c\uc774 \\(M,\\) \\(M &#8216; \\)\uc774 \ub418\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(M &#8216; = P^{-1}MP\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(V=K^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(M\\)\uc740 \ud45c\uc900\uae30\uc800\uc5d0 \ub300\ud55c \\(T_M\\)\uc758 \ud45c\ud604\ud589\ub82c\uc774\ub2e4. \uc774\uc81c \\(P\\)\uc758 \uc5f4\ubca1\ud130\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae30\uc800\uc6d0\uc18c \\(P^1 ,\\) \\(\\cdots ,\\) \\(P^n\\)\uc744 \uc0dd\uac01\ud558\uc790. \uc774 \uae30\uc800\ub85c\ubd80\ud130 \ud45c\uc900\uae30\uc800\ub85c\uc758 \uae30\uc800\ubcc0\ud658\ud589\ub82c\uc740 \\(P\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uae30\uc800\uc6d0\uc18c \\(P^j\\)\uc5d0 \ub300\ud55c \\(T_M\\)\uc758 \ud45c\ud604\ud589\ub82c\uc740 \\(M &#8216; =P^{-1}MP\\)\uc640 \uac19\ub2e4.<\/p>\n<p>\uc5ed\uc774 \uc131\ub9bd\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc740 \uc815\ub9ac 9\ub85c\ubd80\ud130 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n<h2 class=\"margintop2\"><\/h2>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uccb4 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \ucc28\uc6d0\uc774 \\(n\\)\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc740 \\(K^n\\)\uc640 \ub3d9\ud615\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \uc801\ub2f9\ud55c \\(n,\\) \\(m\\)\uc5d0 \ub300\ud558\uc5ec \uc120\ud615\ubcc0\ud658 \\(T : K^n \\rightarrow K^m\\)\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \ub354\uc6b1\uc774 \\(T : K^n \\rightarrow K^m\\)\uc740 \ud589\ub82c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\ubbc0\ub85c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0 \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \ud589\ub82c\uacfc \ub3d9\uc77c\uc2dc\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7f0\ub370 \uc774\ub7ec\ud55c \ud589\ub82c \ud45c\ud604\uc740 \ubca1\ud130\uacf5\uac04\uc5d0 \uc5b4\ub5a0\ud55c \uae30\uc800\uac00 \uc8fc\uc5b4\uc84c\ub294\uc9c0\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \\( \\newcommand{\\Hom}{{\\operatorname{Hom}}} \\newcommand{\\Mat}{{\\operatorname{Mat}}}&hellip;<\/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":[481,438,440,428,434,439],"class_list":["post-5497","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-481","tag-438","tag-440","tag-428","tag-434","tag-439"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5497","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=5497"}],"version-history":[{"count":88,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5497\/revisions"}],"predecessor-version":[{"id":5720,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5497\/revisions\/5720"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}