{"id":9025,"date":"2024-03-20T09:58:24","date_gmt":"2024-03-20T00:58:24","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=9025"},"modified":"2024-04-19T16:33:43","modified_gmt":"2024-04-19T07:33:43","slug":"classification-of-vector-spaces","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/classification-of-vector-spaces\/","title":{"rendered":"\ubca1\ud130\uacf5\uac04\uc758 \ubd84\ub958"},"content":{"rendered":"
\n\\[
\n\\newcommand{\\vecu}{\\mathrm{\\bf{u}}}
\n\\newcommand{\\vecv}{\\mathrm{\\bf{v}}}
\n\\newcommand{\\vecw}{\\mathrm{\\bf{w}}}
\n\\newcommand{\\vecx}{\\mathrm{\\bf{x}}}
\n\\newcommand{\\vecy}{\\mathrm{\\bf{y}}}
\n\\newcommand{\\vecz}{\\mathrm{\\bf{z}}}
\n\\newcommand{\\veczero}{\\mathrm{\\bf{0}}}
\n\\]\n<\/div>\n

\ub3d9\ud615\uc758 \uc758\ubbf8<\/h4>\n

\ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569 \\(\\mathbb{Z}_3\\)\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790.
\n\\[\\mathbb{Z}_3 = \\left\\{ 0 ,\\,\\, 1,\\,\\,2 \\right\\}\\]
\n\uc774 \uc9d1\ud569 \uc704\uc5d0\uc11c \ub367\uc148\uacfc \uacf1\uc148\uc744 \uc0c8\ub86d\uac8c \uc815\uc758\ud574 \ubcf4\uc790. \uc989 \\(m\\)\uacfc \\(n\\)\uc774 \\(\\mathbb{Z}_3\\)\uc758 \uc6d0\uc18c\uc77c \ub54c, \uc0c8\ub85c\uc6b4 \ud569 \\(m+n\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \ud569\uc744 \\(3\\)\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c, \uc0c8\ub85c\uc6b4 \uacf1 \\(mn\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \uacf1\uc744 \\(3\\)\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c \uc815\uc758\ud558\uc790. \uc774\uc640 \uac19\uc740 \ud569\uacfc \uacf1\uc744 \ud45c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c \uadf8\ub9bc\uacfc \uac19\ub2e4.<\/p>\n

\n\"\"<\/a>\n<\/div>\n

\uc774\uc640 \uac19\uc774 \uc0c8\ub86d\uac8c \uc815\uc758\ub41c \ud569\uacfc \uacf1\uc744 \uac00\uc9c4 \uc9d1\ud569 \\(\\mathbb{Z}_3\\)\uc740 \uccb4(field)\uc758 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc989 \\(\\mathbb{Z}_3\\)\ub294 \uccb4\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.
\n\\[F = \\left\\{ 0 ,\\,\\, 4 ,\\,\\, 8 \\right\\}\\]
\n\uc55e\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc774 \uc9d1\ud569\uc5d0 \ub367\uc148\uacfc \uacf1\uc148\uc744 \uc0c8\ub86d\uac8c \uc815\uc758\ud574 \ubcf4\uc790. \uc989 \\(m\\)\uacfc \\(n\\)\uc774 \\(F\\)\uc758 \uc6d0\uc18c\uc77c \ub54c, \uc0c8\ub85c\uc6b4 \ud569 \\(m+n\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \ud569\uc744 \\(12\\)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c, \uc0c8\ub85c\uc6b4 \uacf1 \\(mn\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \uacf1\uc744 \\(12\\)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c \uc815\uc758\ud558\uc790. \uc774\uc640 \uac19\uc740 \ud569\uacfc \uacf1\uc744 \ud45c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c \uadf8\ub9bc\uacfc \uac19\ub2e4.<\/p>\n

\n\"\"<\/a>\n<\/div>\n

\uc774\uc640 \uac19\uc774 \uc0c8\ub86d\uac8c \uc815\uc758\ub41c \ud569\uacfc \uacf1\uc744 \uac00\uc9c4 \uc9d1\ud569 \\(F\\)\ub294 \uccb4\uc774\ub2e4. \uadf8\ub7f0\ub370 \uadf8\ub9bc 1\uacfc \uadf8\ub9bc 2\ub97c \uc798 \uc0b4\ud3b4\ubcf4\uba74, \uc6d0\uc18c\uc758 \uc774\ub984\ub9cc \ub2e4\ub97c \ubfd0 \uac01 \uc9d1\ud569\uc5d0\uc11c \uc0c8\ub86d\uac8c \uc815\uc758\ud55c \ub367\uc148\uacfc \uacf1\uc148\uc5d0 \uc758\ud558\uc5ec \uad6c\uc131\ub418\ub294 \uc6d0\uc18c\ub4e4 \uc0ac\uc774\uc758 \uad00\uacc4\uac00 \uc77c\uce58\ud568\uc744 \uc54c \uc218 \uc788\ub2e4. \uc989 \ud568\uc218 \\(\\phi : \\mathbb{Z} _3 \\rightarrow F \\)\ub97c
\n\\[\\begin{aligned}
\n\\phi ( 0 ) &= 0 , \\\\[6pt]
\n\\phi ( 1 ) &= 4 , \\\\[6pt]
\n\\phi ( 2 ) &= 8
\n\\end{aligned}\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(\\phi\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\uba70, \\(\\mathbb{Z} _3\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(m,\\) \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
\n\\[\\begin{aligned}
\n\\phi (m+n) &= \\phi (m) + \\phi (n) ,\\\\[6pt]
\n\\phi (mn) &= \\phi (m) \\, \\phi (n).
\n\\end{aligned}\\]
\n\uc704 \ub450 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc758 \uad04\ud638 \uc548\uc5d0 \uc788\ub294 \ub367\uc148\uacfc \uacf1\uc148\uc740 \\(\\mathbb{Z} _3\\)\uc5d0 \uc815\uc758\ub41c \uc5f0\uc0b0\uc774\uba70, \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc5d0 \uc788\ub294 \ub367\uc148\uacfc \uacf1\uc148\uc740 \\(F\\)\uc5d0 \uc815\uc758\ub41c \uc5f0\uc0b0\uc774\ub2e4. \uc989 \\(\\phi\\)\ub294 \\(\\mathbb{Z} _3\\)\uc758 \uc6d0\uc18c\ub97c \\(F\\)\uc758 \uc6d0\uc18c\uc5d0 \ud558\ub098\uc529 \ub300\uc751\uc2dc\ud0a4\uba70, \\(\\mathbb{Z} _3\\)\uc758 \uad6c\uc870(\ub367\uc148\uacfc \uacf1\uc148\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \uc6d0\uc18c\ub4e4 \uc0ac\uc774\uc758 \uad00\uacc4)\ub97c \\(F\\)\uc758 \uad6c\uc870\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(\\phi\\)\ub294 \\(\\mathbb{Z} _3\\)\ub85c\ubd80\ud130 \\(F\\)\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1<\/span>(isomorphism)\uc774\uba70, \ub450 \uccb4 \\(\\mathbb{Z} _3\\)\uc640 \\(F\\)\ub294 \uc11c\ub85c \ub3d9\ud615<\/span>(isomorphic)\uc774\ub2e4.<\/p>\n

\ub450 \uc9d1\ud569 \\(\\mathbb{Z} _3\\)\uc640 \\(F\\)\uac00 \ub2e4\ub978 \uc6d0\uc18c\ub97c \uac00\uc9c0\uace0 \uc788\uc73c\ubbc0\ub85c, \ub450 \uc9d1\ud569\uc740 \uac19\uc740 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ub098 \ub450 \uc9d1\ud569\uc774 \uac00\uc9c4 \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \uac19\uace0, \ub450 \uc9d1\ud569\uc774 \uac00\uc9c0\uace0 \uc788\ub294 \uad6c\uc870\uac00 \ub3d9\uc77c\ud558\ubbc0\ub85c, \u201c\uccb4\uc758 \uc131\uc9c8\uc744 \uc5f0\uad6c\ud55c\ub2e4\u201d\ub77c\ub294 \uad00\uc810\uc5d0\uc11c\ub294 \ub450 \uccb4\ub97c \uac19\uc740 \uac83\uc73c\ub85c \ubcf4\uc544\ub3c4 \ubb34\ubc29\ud558\ub2e4.<\/p>\n

\ubca1\ud130\uacf5\uac04\uc758 \ubd84\ub958<\/h4>\n

\uc774\ubc88\uc5d0\ub294 \ubca1\ud130\uacf5\uac04\uc758 \ub3d9\ud615\uc131\uc744 \uc815\uc758\ud574 \ubcf4\uc790. \u201c\\(V\\)\uac00 \uccb4 \\(F\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud560 \ub54c\ub294 \ub124 \uac00\uc9c0, \uc989 \uc2a4\uce7c\ub77c\uc758 \uc9d1\ud569 \\(F,\\) \ubca1\ud130\ub4e4\uc758 \uc9d1\ud569 \\(V,\\) \ubca1\ud130\uc758 \ud569, \uc2a4\uce7c\ub77c \uacf1\uc744 \uc5fc\ub450\uc5d0 \ub450\ub294 \uac83\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \u201c\ub450 \ubca1\ud130\uacf5\uac04\uc774 \uc11c\ub85c \ub3d9\ud615\uc774\ub2e4\u201d\ub77c\ub294 \ud45c\ud604\uc774 \uc774\uc640 \uac19\uc740 \ub124 \uac00\uc9c0 \uad6c\uc131 \uc694\uc18c\uac00 \uc11c\ub85c \uc77c\uce58\ud568\uc744 \uc124\uba85\ud560 \uc218 \uc788\ub3c4\ub85d \ub3d9\ud615\uc131\uc744 \uc815\uc758\ud574\uc57c \ud55c\ub2e4.<\/p>\n

\n

\uc815\uc758. (\ubca1\ud130\uacf5\uac04\uc758 \ub3d9\ud615\uc131)<\/span><\/p>\n

\\(V\\)\uc640 \\(W\\)\uac00 \ub3d9\uc77c\ud55c \uccb4 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(V\\)\uc640 \\(W\\) \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c\ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658 \\(T : V \\rightarrow W\\)\uac00 \uc874\uc7ac\ud558\uba74 \u201c\\(V\\)\uc640 \\(W\\)\ub294 \uc11c\ub85c \ub3d9\ud615<\/span>\uc774\ub2e4(isomorphic)\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uc640 \uac19\uc740 \ud568\uc218 \\(T\\)\ub97c \\(V\\)\ub85c\ubd80\ud130 \\(W\\)\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1<\/span>(isomorphism)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ub450 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc640 \\(W\\)\uac00 \uc11c\ub85c \ub3d9\ud615\uc778 \uac83\uc744 \uae30\ud638\ub85c \u2018\\(V \\cong W\\)\u2019\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n

\ub3d9\ud615\uc778 \ubca1\ud130\uacf5\uac04\uc758 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\mathbb{R}^4\\)\uac00 4\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \ubca1\ud130\uacf5\uac04\uc774\uace0, \\(\\mathrm{Mat}_{2 \\times 2} (\\mathbb{R})\\)\uac00 \\(2\\times 2\\)\uc778 \uc815\uc0ac\uac01\ud615 \uc2e4\ud589\ub82c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

\ub9cc\uc57d \ud568\uc218 \\(\\phi: \\mathbb{R}^4 \\rightarrow \\mathrm{Mat}_{2 \\times 2} (\\mathbb{R})\\)\ub97c
\n\\[\\phi : ( a_1 ,\\,\\, a_2 ,\\,\\, a_3 ,\\,\\, a_4 ) \\,\\mapsto \\,
\n\\left[ \\begin{array}{cc} a_1 & a_2 \\\\ a_3 & a_4 \\end{array} \\right] \\]
\n\uc640 \uac19\uc774 \uc815\uc758\ud558\uba74, \\(\\phi\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658\uc774 \ub41c\ub2e4. \uc989 \\(\\phi\\)\ub294 \ub3d9\ud615\uc0ac\uc0c1\uc774\uace0, \ub450 \ubca1\ud130\uacf5\uac04 \\(\\mathbb{R}^4\\)\uc640 \\(\\mathrm{Mat}_{2 \\times 2} (\\mathbb{R})\\)\ub294 \uc11c\ub85c \ub3d9\ud615\uc774\ub2e4.<\/p>\n

\uc77c\ubc18\uc801\uc73c\ub85c \ub450 \ubca1\ud130\uacf5\uac04\uc774 \ub3d9\ud615\uc778\uc9c0 \uc5ec\ubd80\ub294 \ub450 \ubca1\ud130\uacf5\uac04\uc758 \ucc28\uc6d0\uc744 \ube44\uad50\ud558\uba74 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac. (\ubca1\ud130\uacf5\uac04\uc758 \ubd84\ub958)<\/span><\/p>\n

\\(V\\)\uc640 \\(W\\)\uac00 \ub3d9\uc77c\ud55c \uccb4 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(V\\)\uc640 \\(W\\)\uac00 \uc11c\ub85c \ub3d9\ud615\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(V\\)\uc758 \ucc28\uc6d0\uacfc \\(W\\)\uc758 \ucc28\uc6d0\uc774 \uac19\uc740 \uac83\uc774\ub2e4. \uc989
\n\\[V \\cong W \\quad \\Longleftrightarrow \\quad \\dim (V) = \\dim (W) \\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\uc815\ub9ac\ub97c \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n

\\(V \\cong W \\,\\,\\Rightarrow \\,\\, \\dim(V) = \\dim(W)\\)\uc758 \uc99d\uba85<\/h5>\n

\uc6b0\uc120 \\(V \\cong W\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \ub3d9\ud615\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \uc77c\ub300\uc77c\ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658
\n\\[\\phi : V \\rightarrow W\\]
\n\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \ubca1\ud130\uacf5\uac04\uc740 \uae30\uc800\ub97c \uac00\uc9c0\ubbc0\ub85c \\(V\\)\uc758 \uae30\uc800\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(B\\)\ub97c \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uc9d1\ud569 \\(B ‘ \\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[B’ = \\left\\{ \\phi (\\vecv ) \\,\\vert\\, \\vecv \\in V \\right\\} .\\]
\n\\(\\phi\\)\uac00 \uc77c\ub300\uc77c\ud568\uc218\uc774\ubbc0\ub85c \\(B ‘ \\)\uc740 \\(B\\)\uc640 \uac19\uc740 \uc218\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4. \uc774\uc81c \uc6b0\ub9ac\uc758 \ubaa9\ud45c\ub294 \\(B ‘ \\)\uc774 \\(W\\)\uc758 \uae30\uc800\uac00 \ub428\uc744 \ubcf4\uc774\ub294 \uac83\uc774\ub2e4.<\/p>\n

\\(B ‘ \\)\uc5d0 \uc18d\ud558\ub294 \uc720\ud55c \uac1c\uc758 \ubca1\ud130
\n\\[\\vecw _1 ,\\,\\, \\vecw_2 ,\\,\\, \\cdots ,\\,\\, \\vecw_k \\]
\n\ub97c \uc0dd\uac01\ud558\uc790. \uadf8\ub9ac\uace0 \uc774 \ubca1\ud130\uc758 \uc77c\ucc28\uacb0\ud569\uc774 \\(\\veczero\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790. \uc989 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc0dd\uac01\ud558\uc790.
\n\\[\\alpha_1 \\,\\vecw_1 + \\alpha_2 \\,\\vecw_2 + \\cdots + \\alpha_k \\, \\vecw_k = \\veczero.\\]
\n\uc5ec\uae30\uc11c \uac01 \\(\\alpha_j \\)\ub294 \uc2a4\uce7c\ub77c\uc774\ub2e4. \uc88c\ubcc0\uc758 \uac01 \\(\\vecw_j\\)\ub294 \\(B ‘ \\)\uc758 \uc6d0\uc18c\uc774\ubbc0\ub85c, \\(\\phi(\\vecv _j ) = \\vecw_j \\)\uc778 \\(\\vecv_j\\)\uac00 \\(B\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc989 \uc704 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.
\n\\[\\alpha_1 \\,\\phi( \\vecv_1 ) + \\alpha_2 \\,\\phi( \\vecv_2 ) + \\cdots + \\alpha_k \\, \\phi( \\vecv_k ) = \\veczero.\\]
\n\\(\\phi\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.
\n\\[\\phi ( \\alpha_1 \\, \\vecv_1 + \\alpha_2 \\, \\vecv_2 + \\cdots + \\alpha_k \\,\\vecv_k ) = \\veczero .\\]
\n\\(\\phi\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\alpha_1 \\, \\vecv_1 + \\alpha_2 \\, \\vecv_2 + \\cdots + \\alpha_k \\,\\vecv_k = \\veczero .\\]
\n\uadf8\ub7f0\ub370 \\(\\vecv_1 ,\\) \\(\\vecv_2 ,\\) \\(\\cdots ,\\) \\(\\vecv_k\\)\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c
\n\\[\\alpha_1 = \\alpha_2 = \\cdots = \\alpha_k = 0\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(B ‘ \\)\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(B ‘\\)\uc774 \\(W\\)\ub97c \uc0dd\uc131\ud568\uc744 \ubcf4\uc774\uc790. \\(W\\)\uc5d0 \uc18d\ud558\ub294 \uc784\uc758\uc758 \ubca1\ud130 \\(\\vecw\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\phi\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\ubbc0\ub85c \\(\\phi (\\vecv ) = \\vecw\\)\uc778 \ubca1\ud130 \\(\\vecv\\)\uac00 \\(V\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(B\\)\uac00 \\(V\\)\uc758 \uae30\uc800\uc774\ubbc0\ub85c, \\(B\\)\uc758 \uc6d0\uc18c \\(\\vecv_1 ,\\) \\(\\vecv_2 ,\\) \\(\\cdots ,\\) \\(\\vecv_ p\\)\uc640 \uc2a4\uce7c\ub77c \\(\\beta_1 ,\\) \\(\\beta_2 ,\\) \\(\\cdots ,\\) \\(\\beta_p \\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
\n\\[\\vecv = \\beta_1 \\, \\vecv_1 + \\beta_2 \\,\\vecv_2 + \\cdots + \\beta_p \\,\\vecv_p .\\]
\n\\(j=1,\\,\\,2,\\,\\,\\cdots,\\,\\,p\\)\uc5d0 \ub300\ud558\uc5ec \\(\\vecw_j = \\phi(\\vecv_j )\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\vecw_j\\)\ub294 \\(B ‘ \\)\uc758 \uc6d0\uc18c\uc774\ub2e4. \ub610\ud55c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{aligned}
\n\\vecw
\n&= \\phi( \\vecv ) \\\\[6pt]
\n&= \\phi( \\beta_1 \\, \\vecv_1 + \\beta_2 \\,\\vecv_2 + \\cdots + \\beta_p \\,\\vecv_p ) \\\\[6pt]
\n&= \\beta_1 \\, \\phi( \\vecv_1 ) + \\beta_2 \\, \\phi ( \\vecv_2 ) + \\cdots + \\beta_p \\, \\phi ( \\vecv_p ) \\\\[6pt]
\n&= \\beta_1 \\, \\vecw_1 + \\beta_2 \\,\\vecw_2 + \\cdots + \\beta_p \\,\\vecw_p .
\n\\end{aligned}\\]
\n\uc989 \\(W\\)\uc5d0 \uc18d\ud558\ub294 \uc784\uc758\uc758 \ubca1\ud130\ub294 \\(B ‘ \\)\uc758 \ubca1\ud130\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(B ‘ \\)\uc740 \\(W\\)\ub97c \uc0dd\uc131\ud55c\ub2e4.<\/p>\n

\uacb0\ub860\uc801\uc73c\ub85c \\(B ‘ \\)\uc740 \\(W\\)\uc758 \uae30\uc800\uc774\ub2e4. \\(V\\)\uc758 \uae30\uc800\uc640 \\(W\\)\uc758 \uae30\uc800\uac00 \uac19\uc740 \uc218\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c0\ubbc0\ub85c, \\(V\\)\uc758 \ucc28\uc6d0\uacfc \\(W\\)\uc758 \ucc28\uc6d0\uc774 \uac19\ub2e4.<\/p>\n

\\(\\dim(V) = \\dim(W) \\,\\,\\Rightarrow \\,\\, V \\cong W\\)\uc758 \uc99d\uba85<\/h5>\n

\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \uc989 \\(V\\)\uc640 \\(W\\)\uc758 \ucc28\uc6d0\uc774 \uac19\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(B\\)\uc640 \\(B ‘ \\)\uc744 \uac01\uac01 \\(V\\)\uc640 \\(W\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(B\\)\uc640 \\(B’\\)\uc740 \uc6d0\uc18c\uc758 \uc218\uac00 \uac19\uc740 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc77c\ub300\uc77c\ub300\uc751 \\(T : B \\rightarrow B ‘ \\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(V\\)\uc758 \uae30\uc800\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \\(T\\)\uc758 \ud568\uc22b\uac12\uc774 \uc815\uc758\ub418\uc5c8\uc73c\ubbc0\ub85c, \uc120\ud615 \ud655\uc7a5 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(T\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(V\\)\ub85c \ud655\uc7a5\ud558\uc5ec \uc120\ud615\ubcc0\ud658 \\(T : V \\rightarrow W\\)\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc774\ub807\uac8c \ub9cc\ub4e0 \uc120\ud615\ubcc0\ud658 \\(T\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc784\uc744 \ubcf4\uc774\uba74 \\(T\\)\ub294 \ub3d9\ud615\uc0ac\uc0c1\uc774 \ub41c\ub2e4.<\/p>\n

\ud45c\uae30\ub97c \ud3b8\ud558\uac8c \ud558\uae30 \uc704\ud574
\n\\[\\begin{aligned}
\nB &= \\left\\{ \\vecv_j \\,\\vert\\, j\\in J \\right\\} , \\\\[6pt]
\n\\vecw_j &= \\phi (\\vecv_j ) \\text{ for } j \\in J, \\\\[6pt]
\nB ‘ &= \\left\\{ \\vecw_j \\,\\vert\\, j\\in J \\right\\}
\n\\end{aligned}\\]
\n\ub77c\uace0 \ud558\uc790.<\/p>\n

\\(T\\)\uac00 \uc77c\ub300\uc77c\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(\\vecu\\)\uc640 \\(\\vecv\\)\uac00 \\(V\\)\uc758 \ubca1\ud130\uc774\uace0 \\(\\vecu \\ne \\vecv \\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(\\vecu\\)\uc640 \\(\\vecv\\)\ub97c \\(B\\)\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ud558\ub294 \ub370\uc5d0 \uc0ac\uc6a9\ub418\ub294 \\(B\\)\uc758 \uc6d0\uc18c \uc804\uccb4\ub97c \\(\\vecv_1 ,\\) \\(\\vecv_2 ,\\) \\(\\cdots ,\\) \\(\\vecv_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\vecu\\)\uc640 \\(\\vecv\\)\ub294 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.
\n\\[\\begin{aligned}
\n\\vecu &= \\alpha_1 \\,\\vecv_1 + \\alpha_2 \\,\\vecv_2 + \\cdots + \\alpha_n \\,\\vecv_n , \\\\[6pt]
\n\\vecv &= \\beta_1 \\,\\vecv_1 + \\beta_2 \\,\\vecv_2 + \\cdots + \\beta_n \\,\\vecv_n .
\n\\end{aligned}\\]
\n\uc5ec\uae30\uc11c \ub450 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \\(\\alpha_j\\)\uc640 \\(\\beta_j\\)\ub294 \uc2a4\uce7c\ub77c\uc774\ub2e4. \\(\\phi\\)\uac00 \uc120\ud615\ubcc0\ud658\uc774\ubbc0\ub85c \\(T(\\vecv )\\)\uc640 \\(T(\\vecu )\\)\ub294 \uac01\uac01 \\(B ‘ \\)\uc758 \uc6d0\uc18c\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4:
\n\\[\\begin{aligned}
\nT(\\vecv ) &= T( \\alpha_1 \\,\\vecv_1 + \\alpha_2 \\,\\vecv_2 + \\cdots + \\alpha_n \\,\\vecv_n ) \\\\[6pt]
\n&= \\alpha_1 \\, T( \\vecv_1 ) + \\alpha_2 \\, T( \\vecv_2 ) + \\cdots + \\alpha_n \\, T( \\vecv_n ) \\\\[6pt]
\n&= \\alpha_1 \\,\\vecw_1 + \\alpha_2 \\,\\vecw_2 + \\cdots + \\alpha_n \\,\\vecw_n \\,, \\\\[10pt]<\/p>\n

T(\\vecu ) &= T( \\beta_1 \\,\\vecu_1 + \\beta_2 \\,\\vecu_2 + \\cdots + \\beta_n \\,\\vecu_n ) \\\\[6pt]
\n&= \\beta_1 \\, T( \\vecu_1 ) + \\beta_2 \\, T( \\vecu_2 ) + \\cdots + \\beta_n \\, T( \\vecu_n ) \\\\[6pt]
\n&= \\beta_1 \\,\\vecw_1 + \\beta_2 \\,\\vecw_2 + \\cdots + \\beta_n \\,\\vecw_n \\,.
\n\\end{aligned}\\]
\n\ud55c \ubca1\ud130\ub97c \uae30\uc800\uc758 \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ud558\ub294 \ud615\ud0dc\ub294 \uc720\uc77c\ud558\ud558\ub2e4. \uadf8\ub7f0\ub370 \\(\\vecu \\ne \\vecv \\)\uc774\ubbc0\ub85c \\(\\alpha_p \\ne \\beta_p\\)\uc778 \\(p\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(T(\\vecv ) \\ne T(\\vecu )\\)\uc774\ub2e4. \uc989 \\(T\\)\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(T\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(W\\)\uc758 \ubca1\ud130 \\(\\vecw\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(B ‘ \\)\uc774 \\(W\\)\uc758 \uae30\uc800\uc774\ubbc0\ub85c,
\n\\[\\vecw = \\lambda_1 \\,\\vecw_1 + \\lambda_2 \\, \\vecw_2 + \\cdots + \\lambda_q \\, \\vecw_q\\]
\n\uc778 \\(B ‘\\)\uc758 \uc6d0\uc18c \\(\\vecw_1 ,\\) \\(\\vecw_2 ,\\) \\(\\cdots ,\\) \\(\\vecw_q \\)\uc640 \uc2a4\uce7c\ub77c \\(\\lambda_1 ,\\) \\(\\lambda_2 ,\\) \\(\\cdots ,\\) \\(\\lambda_q\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(V\\)\uc758 \ubca1\ud130 \\(\\vecv\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[\\vecv = \\lambda_1 \\,\\vecv_1 + \\lambda_2 \\, \\vecv_2 + \\cdots + \\lambda_q \\, \\vecv_q\\]
\n\uadf8\ub7ec\uba74 \\(T\\)\uc758 \uc120\ud615\uc131\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{aligned}
\nT(\\vecv)
\n&= T( \\lambda_1 \\,\\vecv_1 + \\lambda_2 \\, \\vecv_2 + \\cdots + \\lambda_q \\, \\vecv_q ) \\\\[6pt]
\n&= \\lambda_1 \\, T( \\vecv_1 ) + \\lambda_2 \\, T( \\vecv_2 ) + \\cdots + \\lambda_q \\, T( \\vecv_q ) \\\\[6pt]
\n&= \\lambda_1 \\,\\vecw_1 + \\lambda_2 \\, \\vecw_2 + \\cdots + \\lambda_q \\, \\vecw_q \\\\[6pt]
\n&= \\vecw .
\n\\end{aligned}\\]
\n\uc989 \\(T(\\vecv) = \\vecw\\)\uc778 \ubca1\ud130 \\(\\vecv\\)\uac00 \\(V\\)\uc5d0 \uc874\uc7ac\ud558\ubbc0\ub85c, \\(\\vecw\\)\ub294 \\(T\\)\uc758 \uce58\uc5ed\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(T\\)\ub294 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/p>\n

\uacb0\ub860\uc801\uc73c\ub85c \\(T : V \\rightarrow W\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc778 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub450 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc640 \\(W\\)\ub294 \uc11c\ub85c \ub3d9\ud615\uc774\ub2e4.\n<\/p>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

\\( \\newcommand{\\vecu}{\\mathrm{\\bf{u}}} \\newcommand{\\vecv}{\\mathrm{\\bf{v}}} \\newcommand{\\vecw}{\\mathrm{\\bf{w}}} \\newcommand{\\vecx}{\\mathrm{\\bf{x}}} \\newcommand{\\vecy}{\\mathrm{\\bf{y}}} \\newcommand{\\vecz}{\\mathrm{\\bf{z}}} \\newcommand{\\veczero}{\\mathrm{\\bf{0}}} \\) \ub3d9\ud615\uc758 \uc758\ubbf8 \ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569 \\(\\mathbb{Z}_3\\)\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790. \\(\\mathbb{Z}_3 = \\left\\{ 0 ,\\,\\, 1,\\,\\,2 \\right\\}\\) \uc774 \uc9d1\ud569 \uc704\uc5d0\uc11c \ub367\uc148\uacfc \uacf1\uc148\uc744 \uc0c8\ub86d\uac8c \uc815\uc758\ud574 \ubcf4\uc790. \uc989 \\(m\\)\uacfc \\(n\\)\uc774 \\(\\mathbb{Z}_3\\)\uc758 \uc6d0\uc18c\uc77c \ub54c, \uc0c8\ub85c\uc6b4 \ud569 \\(m+n\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \ud569\uc744 \\(3\\)\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c, \uc0c8\ub85c\uc6b4 \uacf1 \\(mn\\)\uc744 \\(m\\)\uacfc \\(n\\)\uc758 \uacf1\uc744 \\(3\\)\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c \uc815\uc758\ud558\uc790. \uc774\uc640 \uac19\uc740 \ud569\uacfc \uacf1\uc744 \ud45c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c…<\/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":[],"class_list":["post-9025","post","type-post","status-publish","format-standard","hentry","category-linear-algebra"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9025","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=9025"}],"version-history":[{"count":46,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9025\/revisions"}],"predecessor-version":[{"id":9073,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9025\/revisions\/9073"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9025"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=9025"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=9025"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}