{"id":6276,"date":"2021-05-02T19:04:11","date_gmt":"2021-05-02T10:04:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=6276"},"modified":"2021-05-03T22:47:07","modified_gmt":"2021-05-03T13:47:07","slug":"relation-between-linear-transformations-and-matrices-general-spaces","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/relation-between-linear-transformations-and-matrices-general-spaces\/","title":{"rendered":"\uc120\ud615\ubcc0\ud658\uacfc \ud589\ub82c\uc758 \uad00\uacc4 (\uc77c\ubc18\uc801\uc778 \ubca1\ud130\uacf5\uac04)"},"content":{"rendered":"

\uc9c0\ub09c \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \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\uc558\ub2e4(\uc9c0\ub09c \ud3ec\uc2a4\ud305 \ubcf4\uae30<\/a>). \uc774\ubc88\uc5d0\ub294 \uc77c\ubc18\uc801\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V,\\) \\(V’\\) \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uacfc \ud589\ub82c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\\(K\\)\uac00 \uccb4\uc774\uace0 \\(n\\)\uacfc \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uc640 \\(V’\\)\uc774 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04, \\(m\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c
\n\\[\\begin{align}
\nB: &\\,\\, v_1 ,\\, v_2 ,\\, \\cdots ,\\, v_n, \\\\[6pt]
\nB’ : & \\,\\, v_1 ‘ ,\\, v_2 ‘ ,\\, \\cdots ,\\, v_m ‘
\n\\end{align}\\]
\n\uc774 \uac01\uac01 \\(V\\)\uc640 \\(V’\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(v\\in V\\)\ub294 \\(B\\)\uc758 \ubca1\ud130\ub4e4\uc758 \uc77c\ucc28\uacb0\ud569
\n\\[v = k_1 v_1 + k_2 v_2 + \\cdots + k_n v_n\\]
\n\uc73c\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4. \uc774\ub54c \\(v\\)\ub97c \uc21c\uc11c\uc30d \\((k_1 ,\\, k_2 ,\\, \\cdots ,\\, k_n)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218
\n\\[\\begin{gather}
\n\\gamma_B :\\, V \\rightarrow K^n ,\\\\[6pt]
\n\\gamma_B (v) = (k_1 ,\\, k_2 ,\\, \\cdots ,\\, k_n )
\n\\end{gather}\\]
\n\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uac19\uc740 \ud568\uc218 \\(\\gamma_B\\)\ub97c \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uc88c\ud45c\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \ud568\uc22b\uac12 \\(\\gamma_B(v)\\)\ub97c \uae30\uc800 \\(B\\)\uc5d0 \ub300\ud55c \\(v\\)\uc758 \uc88c\ud45c<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\gamma_B\\)\ub294 \\(V\\)\ub85c\ubd80\ud130 \\(K^n\\)\uc73c\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774 \ub41c\ub2e4.<\/p>\n

\ub611\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uae30\uc800 \\(B’\\)\uc5d0 \ub300\ud55c \\(V\\)\uc758 \uc88c\ud45c\ud568\uc218 \\(\\gamma_{B’}\\)\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\uba70, \\(\\gamma_{B’}\\)\uc740 \\(V’\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774 \ub41c\ub2e4.<\/p>\n

\ud568\uc218 \\(T\\)\uac00 \\(V\\)\ub85c\ubd80\ud130 \\(V’\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V\\)\uc758 \ubca1\ud130 \\(v\\)\ub294 \\(T\\)\uc5d0 \uc758\ud558\uc5ec \\(V’\\)\uc758 \ubca1\ud130 \\(T(v)\\)\uc5d0 \ub300\uc751\ub41c\ub2e4. \\(v\\in V\\)\uc640 \\(T(v)\\in V’\\)\uc774 \uac01\uac01 \ud558\ub098\uc529 \uc8fc\uc5b4\uc9c8 \ub54c\ub9c8\ub2e4 \\(\\gamma_B (v)\\)\uc640 \\(\\gamma_{B’} (T(v))\\)\ub3c4 \uac01\uac01 \ud558\ub098\uc529 \uc8fc\uc5b4\uc9c0\uba70, \uc774\ub4e4\uc740 \uac01\uac01 \\(K^n,\\) \\(K^m\\)\uc758 \uc6d0\uc18c\uc774\ub2e4. \uc5ec\uae30\uc11c
\n\\[(\\gamma_{B’} \\circ T \\circ \\gamma_{B}^{-1} )(v) = \\gamma_{B’} (T(v))\\]
\n\uc774\ubbc0\ub85c, \ud569\uc131\ud568\uc218 \\(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1}\\)\ub294 \\(K^n\\)\uc758 \uc6d0\uc18c \\(\\gamma_B (v)\\)\ub97c \\(K^m\\)\uc758 \uc6d0\uc18c \\(\\gamma_{B’} (T(v))\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ub2e4. \ub354\uc6b1\uc774 \\(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1}\\)\ub294 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ud569\uc131\ud568\uc218\uc774\ubbc0\ub85c \uadf8 \uc790\uccb4\ub85c\uc11c \uc120\ud615\ubcc0\ud658\uc774\ub2e4.<\/p>\n

\\(V\\)\ub85c\ubd80\ud130 \\(V’\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ubaa8\uc784\uc744 \\(\\operatorname{Hom}(V,\\,V’)\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uc9d1\ud569\uc5d0 \ud1b5\uc0c1\uc801\uc778 \uc2a4\uce7c\ub77c\uacf1\uacfc \ud568\uc218\uc758 \ud569\uc744 \uac01\uac01 \uc2a4\uce7c\ub77c\uacf1\uacfc \ud568\uc218\uc758 \ud569\uc73c\ub85c \uc815\uc758\ud558\uba74, \uc774 \uc9d1\ud569\uc740 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774 \ub41c\ub2e4.<\/p>\n

\uac01 \\(T\\in\\operatorname{Hom}(V,\\,V’)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1}\\)\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c, \uc774 \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c \\(M(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1})\\)\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774 \ud589\ub82c\uc744 \uac04\ub2e8\ud788 \\(M_{B,B’}(T)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[M_{B,B’}(T) = M(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1})\\]
\n\uc774\ub2e4. \uc774 \ud589\ub82c\uc744 \uae30\uc800 \\(B\\)\uc640 \\(B’\\)\uc5d0 \ub300\ud55c \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc774\uc81c \ub450 \uacf5\uac04 \\(\\operatorname{Hom}(V,\\,V’)\\)\uacfc \\(\\operatorname{Mat}_{m\\times n}(K)\\)\uac00 \ub3d9\ud615\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n

\n

\uc815\ub9ac 1.<\/span>
\n\ub450 \ubca1\ud130\uacf5\uac04 \\(\\operatorname{Hom}(V,\\,V’)\\)\uacfc \\(\\operatorname{Mat}_{m\\times n}(K)\\)\ub294 \ub3d9\ud615\uc774\ub2e4.\n<\/p>\n<\/div>\n

\ub450 \ubca1\ud130\uacf5\uac04\uc774 \ub3d9\ud615\uc784\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \ub450 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0 \uc801\uc808\ud55c \ud568\uc218 \\(\\phi\\)\ub97c \uc815\uc758\ud558\uace0, \uc774 \ud568\uc218\uac00 \ub3d9\ud615\uc0ac\uc0c1\uc784\uc744 \ubcf4\uc77c \uac83\uc774\ub2e4.<\/p>\n

1\ub2e8\uacc4. \ub450 \uacf5\uac04 \uc0ac\uc774\uc758 \ud568\uc218 \\(\\phi\\)\ub97c \uc815\uc758\ud558\uc790.<\/h4>\n

\uc120\ud615\ubcc0\ud658 \\(T: V \\rightarrow V’\\)\uc744 \\(M_{B,B’}(T)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218
\n\\[\\begin{gather}
\n\\phi : \\, \\operatorname{Hom}(V,\\,V’) \\rightarrow \\operatorname{Mat}_{m\\times n}(K) ,\\\\[6pt]
\n\\phi : \\, T \\mapsto M_{B,B’} (T)
\n\\end{gather}\\]
\n\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \ud568\uc218\uac00 \ubc14\ub85c \uc6b0\ub9ac\uac00 \ubc14\ub77c\ub294 \ub3d9\ud615\uc0ac\uc0c1\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n

2\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/h4>\n

\uac01 \uc120\ud615\ubcc0\ud658 \\(T: V \\rightarrow V’\\)\uc5d0 \ub300\ud558\uc5ec \\(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1}\\)\ub294 \\(K^n\\)\uc73c\ub85c\ubd80\ud130 \\(K^m\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c, \uc774 \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c
\n\\[M_{B,B’}(T) = M(\\gamma_{B’} \\circ T \\circ \\gamma_B ^{-1})\\]
\n\\(m\\times n\\) \ud589\ub82c\ub85c\uc11c \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub41c\ub2e4. \uc989 \\(\\phi\\)\ub294 \\(\\operatorname{Hom}(V,\\,V’)\\)\uc758 \uc6d0\uc18c\ub97c \\(\\operatorname{Mat}_{m\\times n}(K)\\)\uc758 \uc6d0\uc18c\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\phi\\)\ub294 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/p>\n

3\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub2e4.<\/h4>\n

\uc2a4\uce7c\ub77c \\(k\\in K\\)\uc640 \uc120\ud615\ubcc0\ud658 \\(T:V\\rightarrow V’\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(kT\\) \ub610\ud55c \\(V\\)\ub85c\ubd80\ud130 \\(V’\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c \\(M_{B,B’}(kT)\\)\ub294 \\(m\\times n\\) \ud589\ub82c\uc774\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(\\mathbf{x}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\begin{align}
\n(M_{B,B’}(kT))\\mathbf{x}
\n&= (\\gamma_{B’}\\circ (kT) \\circ \\gamma_B ^{-1})(\\mathbf{x}) \\\\[6pt]
\n&= k(\\gamma_{B’} \\circ T \\circ \\gamma_B^{-1})(\\mathbf{x})) \\\\[6pt]
\n&= k((M_{B,B’} (T))\\mathbf{x}) \\\\[6pt]
\n&= (k(M_{B,B’} (T)))\\mathbf{x}
\n\\end{align}\\]
\n\uc774\ub2e4. \uc989 \\(K^n\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(\\mathbf{x}\\)\uc5d0 \ub300\ud558\uc5ec \\((M_{B,B’}(kT))\\mathbf{x}\\)\uc640 \\((k(M_{B,B’} (T)))\\mathbf{x}\\)\uac00 \uc77c\uce58\ud558\ubbc0\ub85c, \ub450 \ud589\ub82c \\(M_{B,B’}(kT)\\)\uc640 \\(k(M_{B,B’} (T))\\)\ub294 \uac19\uc740 \ud589\ub82c\uc774\ub2e4. \uc989
\n\\[\\phi(kT) = M_{B,B’}(kT) = k(M_{B,B’} (T)) = k\\phi(T)\\]
\n\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \ub450 \uc120\ud615\ubcc0\ud658 \\(T_1 : V \\rightarrow V’,\\) \\(T_2 : V \\rightarrow V’\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(T_1 +T_2\\) \ub610\ud55c \\(V\\)\ub85c\ubd80\ud130 \\(V’\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c \\(M_{B,B’}(T_1 + T_2 )\\)\ub294 \\(m\\times n\\) \ud589\ub82c\uc774\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(\\mathbf{x}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\begin{align}
\n(M_{B,B’}(T_1 +T_2))\\mathbf{x}
\n&= (\\gamma_{B’} \\circ (T_1 + T_2 ) \\circ \\gamma_B ^{-1})(\\mathbf{x}) \\\\[6pt]
\n&= \\gamma_{B’} ((T_1 +T_2 )(\\gamma_B ^{-1} (\\mathbf{x})) \\\\[6pt]
\n&= \\gamma_{B’} (T_1 (\\gamma_B^{-1} (\\mathbf{x})) + T_2 ( \\gamma_B ^{-1} (\\mathbf{x})) \\\\[6pt]
\n&= \\gamma_{B’} (T_1 (\\gamma_B^{-1}(\\mathbf{x}))) + \\gamma_{B’} (T_2 (\\gamma_B ^{-1} (\\mathbf{x}))) \\\\[6pt]
\n&= (M_{B,B’} (T_1))\\mathbf{x} + (M_{B,B’}(T_2))\\mathbf{x} \\\\[6pt]
\n&= (M_{B,B’} (T_1) + M_{B,B’}(T_2))\\mathbf{x}
\n\\end{align}\\]
\n\uc774\ub2e4. \uc989 \\(K^n\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(\\mathbf{x}\\)\uc5d0 \ub300\ud558\uc5ec \\((M_{B,B’}(T_1 +T_2))\\mathbf{x}\\)\uc640 \\((M_{B,B’} (T_1) + M_{B,B’}(T_2))\\mathbf{x}\\)\uac00 \uc77c\uce58\ud558\ubbc0\ub85c, \ub450 \ud589\ub82c \\(M_{B,B’}(T_1 +T_2)\\)\uc640 \\(M_{B,B’} (T_1) + M_{B,B’}(T_2)\\)\ub294 \uac19\uc740 \ud589\ub82c\uc774\ub2e4. \uc989
\n\\[\\phi(T_1 + T_2 ) = M_{B,B’}(T_1 + T_2 ) = M_{B,B’}(T_1 ) + M_{B,B’}(T_2) = \\phi(T_1) +\\phi(T_2)\\]
\n\uc774\ub2e4. \uc774\ub85c\uc368 \\(\\phi\\)\uac00 \uc120\ud615\uc784\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n

4\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/h4>\n

\uc11c\ub85c \ub2e4\ub978 \ub450 \uc120\ud615\ubcc0\ud658 \\(T_1 : V \\rightarrow V’ ,\\) \\(T_2 : V \\rightarrow V’\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(T_1\\)\uacfc \\(T_2\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c \\(V\\)\uc758 \uae30\uc800\uc6d0\uc18c \uc911\uc5d0 \ub450 \ud568\uc218\uc758 \ud568\uc22b\uac12\uc774 \ub2ec\ub77c\uc9c0\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(v_j\\)\uac00 \\(V\\)\uc758 \uae30\uc800\uc6d0\uc18c \uc911 \ud558\ub098\uc774\uace0 \\(T_1(v_j) \\ne T_2 (v_j)\\)\ub77c\uace0 \ud558\uc790. \\(\\gamma_B (v_j) = \\mathbf{e}_j\\)\ub294 \\(K^n\\)\uc758 \ud45c\uc900\uae30\uc800\uc6d0\uc18c \uc911 \\(j\\)\uc9f8 \ubca1\ud130\uc774\ub2e4. \uc774\ub54c
\n\\[\\begin{align}
\n(M_{B,B’}(T_1))^j
\n&= (M_{B,B’}(T_1))\\mathbf{e}_j \\\\[6pt]
\n&= (M_{B,B’}(T_1))(\\gamma_B(v_j)) \\\\[6pt]
\n&= (\\gamma_{B’} \\circ T_1 \\circ \\gamma_B ^{-1})(\\gamma_B(v_j)) \\\\[6pt]
\n&= \\gamma_{B’} (T_1 (v_j)) \\\\[6pt]
\n&\\ne \\gamma_{B’} (T_2 (v_j)) \\\\[6pt]
\n&= (\\gamma_{B’} \\circ T_2 \\circ \\gamma_B ^{-1})(\\gamma_B(v_j)) \\\\[6pt]
\n&= (M_{B,B’}(T_2))(\\gamma_B(v_j)) \\\\[6pt]
\n&= (M_{B,B’}(T_2))\\mathbf{e}_j \\\\[6pt]
\n&= (M_{B,B’}(T_2))^j
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c \\(M_{B,B’}(T_1)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uacfc \\(M_{B,B’}(T_2)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc740 \uc11c\ub85c \ub2e4\ub974\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\phi(T_1) = M_{B,B’}(T_1) \\ne M_{B,B’}(T_2) = \\phi(T_2)\\]
\n\uc774\ub2e4. \uc989 \\(\\phi\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/p>\n

5\ub2e8\uacc4. \\(\\phi\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/h4>\n

\\(m\\times n\\) \ud589\ub82c \\(A\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc120\ud615\ubcc0\ud658 \\(T:V\\rightarrow V’\\)\uc744
\n\\[\\forall v\\in V : \\, T(v) = \\gamma_{B’}^{-1} ( T_A ( \\gamma_B (v)))\\]
\n\ub85c \uc815\uc758\ud558\uc790. \\(T\\)\ub294 \uc120\ud615\ubcc0\ud658\ub4e4\uc758 \ud569\uc131\ud568\uc218\uc774\ubbc0\ub85c \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \ub610\ud55c \\(\\gamma_B\\)\uc758 \uc815\uc758\uc5ed\uc774 \\(V\\)\uc774\uace0 \\(\\gamma_{B’}^{-1}\\)\uc758 \uacf5\uc5ed\uc774 \\(V’\\)\uc774\ubbc0\ub85c \\(T\\)\ub294 \\(V\\)\ub85c\ubd80\ud130 \\(V’\\)\uc73c\ub85c\uc758 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc774\ub54c
\n\\[\\begin{align}
\n\\phi(T) &= M_{B,B’}(T) \\\\[6pt]
\n&= M(\\gamma_{B’} \\circ (\\gamma_{B’}^{-1} \\circ T_A \\circ \\gamma_B ) \\circ \\gamma_B ^{-1}) \\\\[6pt]
\n&= M(T_A) \\\\[6pt]
\n&= A
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c \\(\\phi\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/p>\n

\uc774\ub85c\uc368 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V,\\) \\(V’\\)\uc5d0 \uae30\uc800 \\(B,\\) \\(B’\\)\uc774 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c, \uc120\ud615\ubcc0\ud658 \\(T:V\\rightarrow V’\\)\uc740 \ud589\ub82c\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uac04\uc8fc\ud560 \uc218 \uc788\ub2e4. \uadf8\ub807\ub2e4\uba74 \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(M_{B,B’}(T)\\)\ub97c \uc5b4\ub5bb\uac8c \uad6c\ud560 \uac83\uc778\uac00? \uadf8 \ub2f5\uc740 \uc704 \uc815\ub9ac\uc758 \uc99d\uba85 \uacfc\uc815 4\ub2e8\uacc4\uc5d0 \uc788\ub2e4. \\(v_j\\)\uac00 \\(V\\)\uc758 \uae30\uc800\uc758 \\(j\\)\uc9f8 \ubca1\ud130\uc77c \ub54c, \\(M_{B,B’}(T)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc740 \\(\\gamma_{B’}(T(v_j))\\)\uc640 \uac19\ub2e4. \uc989<\/p>\n

\n\\[M_{B,B’}(T) = [ \\gamma_{B’}(T(v_1)) \\,\\,\\, \\gamma_{B’}(T(v_2)) \\,\\,\\, \\cdots \\,\\,\\, \\gamma_{B’}(T(v_n)) ]\\]\n<\/div>\n

\uc774\ub2e4. \uc5ec\uae30\uc11c \uac01 \\(\\gamma_{B’}(T(v_j))\\)\ub294 \\(m\\times 1\\) \ud589\ub82c\ub85c \ub098\ud0c0\ub098\ub294 \uc5f4\ubca1\ud130\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc740 \\(m\\times n\\) \ud589\ub82c\uc744 \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(V=K^n,\\) \\(V’ = K^m\\)\uc774\uba74 \\(\\gamma_B\\)\uc640 \\(\\gamma_{B’}\\)\uc740 \ud56d\ub4f1\ud568\uc218\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc740 \\(T\\)\uc758 \ud45c\ud604\ud589\ub82c \\(M(T)\\)\uc758 \uc815\uc758\uc640 \uc77c\uce58\ud55c\ub2e4.<\/p>\n

\ub05d\uc73c\ub85c \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uc744 \ud589\ub82c\ub85c \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\n

\n\uc815\ub9ac 2.<\/span>
\n\\(V,\\) \\(V’,\\) \\(V”\\)\uc774 \uac01\uac01 \\(p\\)\ucc28\uc6d0, \\(n\\)\ucc28\uc6d0, \\(m\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0, \\(B,\\) \\(B’,\\) \\(B”\\)\uc774 \uac01\uac01 \\(V,\\) \\(V’,\\) \\(V”\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(T:V \\rightarrow V’,\\) \\(T’ : V’ \\rightarrow V”\\)\uc774 \uc120\ud615\ubcc0\ud658\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[M_{B,B”}(T’\\circ T) = M_{B’,B”}(T’) \\cdot M_{B,B’}(T).\\]
\n\uc989 \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\ud568\uc218\uc758 \ud45c\ud604\ud589\ub82c\uc740 \uac01 \uc120\ud615\ubcc0\ud658\uc758 \ud45c\ud604\ud589\ub82c\uc758 \uacf1\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85.<\/p>\n

\\(M_{B’,B”}(T’)\\)\uc740 \\(m\\times n\\) \ud589\ub82c\uc774\uace0 \\(M_{B,B’}(T)\\)\ub294 \\(n\\times p\\) \ud589\ub82c\uc774\ubbc0\ub85c \ub450 \ud589\ub82c\uc758 \uacf1\uc740 \\(m\\times p\\) \ud589\ub82c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc5d0\uc11c \uc591\ubcc0\uc758 \ud589\ub82c\uc758 \ud06c\uae30\ub294 \uc77c\uce58\ud55c\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c, \uac01 \\(\\mathbf{x}\\in K^p\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\begin{align}
\n(M_{B,B”}(T’\\circ T))\\mathbf{x}
\n&= (\\gamma_{B”} \\circ (T’ \\circ T)\\circ \\gamma_B^{-1} )(\\mathbf{x}) \\\\[6pt]
\n&= (\\gamma_{B”} \\circ T’ \\circ \\gamma_{B’}^{-1} \\circ \\gamma_B \\circ T \\circ \\gamma_B^{-1})(\\mathbf{x}) \\\\[6pt]
\n&= (\\gamma_{B”} \\circ T’ \\circ \\gamma_{B’}^{-1})((\\gamma_B \\circ T \\circ \\gamma_B^{-1})(\\mathbf{x})) \\\\[6pt]
\n&= (M_{B’,B”}(T’))((M_{B,B’}(T))\\mathbf{x}) \\\\[6pt]
\n&= ((M_{B’,B”} (T’))(M_{B,B’}(T)))\\mathbf{x}
\n\\end{align}\\]
\n\uc774\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\mathbf{x}\\in K^p\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[(M_{B,B”}(T’\\circ T))\\mathbf{x} = ((M_{B’,B”} (T’))(M_{B,B’}(T)))\\mathbf{x}\\]
\n\uc774\ubbc0\ub85c \ub450 \ud589\ub82c \\(M_{B,B”}(T’\\circ T)\\)\uc640 \\((M_{B’,B”} (T’))(M_{B,B’}(T))\\)\ub294 \uc77c\uce58\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc694\uc57d\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uae30\uc800\uac00 \uc8fc\uc5b4\uc9c4 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uc740 \ud589\ub82c\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\ub54c \uc120\ud615\ubcc0\ud658\uc758 \ud569\uc131\uc740 \ud589\ub82c\uc758 \uacf1\uacfc \uac19\ub2e4.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\uc9c0\ub09c \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \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\uc558\ub2e4(\uc9c0\ub09c \ud3ec\uc2a4\ud305 \ubcf4\uae30). \uc774\ubc88\uc5d0\ub294 \uc77c\ubc18\uc801\uc778 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V,\\) \\(V’\\) \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \uc120\ud615\ubcc0\ud658\uacfc \ud589\ub82c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(K\\)\uac00 \uccb4\uc774\uace0 \\(n\\)\uacfc \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(V\\)\uc640 \\(V’\\)\uc774 \\(K\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \\(n\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04, \\(m\\)\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(\\begin{align} B: &\\,\\, v_1 ,\\, v_2 ,\\, \\cdots ,\\, v_n, B’ : & \\,\\, v_1 ‘ ,\\,…<\/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-6276","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\/6276","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=6276"}],"version-history":[{"count":32,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6276\/revisions"}],"predecessor-version":[{"id":6319,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6276\/revisions\/6319"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=6276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=6276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}