{"id":5401,"date":"2020-10-14T18:45:07","date_gmt":"2020-10-14T09:45:07","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5401"},"modified":"2020-10-14T19:23:50","modified_gmt":"2020-10-14T10:23:50","slug":"linear-transformation-between-vector-spaces-over-q","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-transformation-between-vector-spaces-over-q\/","title":{"rendered":"\uc720\ub9ac\uc218\uccb4 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uacf5\uac04 \uc0ac\uc774\uc758 \uc120\ud615\ubcc0\ud658"},"content":{"rendered":"

\\(V\\)\uc640 \\(W\\)\uac00 \uccb4 \\(F\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(T : V \\rightarrow W\\)\uac00 \ub450 \uc870\uac74<\/p>\n

    \n
  1. \uc784\uc758\uc758 \\(v_1 ,\\, v_2 \\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(T(v_1 + v_2 ) = T(v_1 ) + T(v_2 )\\)\uc774\ub2e4,<\/li>\n
  2. \uc784\uc758\uc758 \\(k \\in F\\)\uc640 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(T(kv) = kT(v)\\)\uc774\ub2e4<\/li>\n<\/ol>\n

    \ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub97c \uc120\ud615\ubcc0\ud658<\/span>(linear transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \ub9cc\uc57d \\(F = \\mathbb{R}\\)\ub77c\uba74, \\(T\\)\uac00 (1)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub354\ub77c\ub3c4 (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\uc744 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \\(F=\\mathbb{Q}\\)\ub77c\uba74 \uc774\uc57c\uae30\uac00 \ub2ec\ub77c\uc9c4\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac.<\/span>
    \n\\(V\\)\uc640 \\(W\\)\uac00 \\(\\mathbb{Q}\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(T : V \\rightarrow W\\)\uac00 \uc784\uc758\uc758 \\(v_1 ,\\, v_2 \\in V\\)\uc5d0 \ub300\ud558\uc5ec \\[T(v_1 + v_2 ) = T(v_1) + T(v_2 )\\]\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub294 \uc120\ud615\ubcc0\ud658\uc774\ub2e4. \uc989 \\(F=\\mathbb{Q}\\)\uc77c \ub54c (1)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 (2)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \ub2e4\uc12f \ub2e8\uacc4\ub85c \uc99d\uba85\ud558\uc790.<\/p>\n

    1\ub2e8\uacc4.<\/span>
    \n\\(T(\\mathbf{0}) = T(\\mathbf{0}+\\mathbf{0})=T(\\mathbf{0}) + T(\\mathbf{0})\\)\uc774\ubbc0\ub85c \\(T(\\mathbf{0})=\\mathbf{0}\\)\uc774\ub2e4.<\/p>\n

    2\ub2e8\uacc4.<\/span>
    \n\uc784\uc758\uc758 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\mathbf{0}=T(\\mathbf{0}) = T(v+(-v)) = T(v) + T(-v)\\]
    \n\uc774\ubbc0\ub85c \\(T(-v) = -T(v)\\)\uc774\ub2e4.<\/p>\n

    3\ub2e8\uacc4.<\/span>
    \n\ubca1\ud130 \\(v\\in V\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[T(nv) = nT(v)\\tag{3}\\]
    \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 \\(n=1\\)\uc77c \ub54c (3)\uc740 \ucc38\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(n=k\\)\uc77c \ub54c (3)\uc774 \ucc38\uc774\ub77c\uace0 \uac00\uc815\ud558\uba74
    \n\\[T((k+1)v) = T(kv + v) = T(kv) + T(v) = kT(v) +T(v) = (k+1)T(v)\\]
    \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec (3)\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n

    4\ub2e8\uacc4.<\/span>
    \n\ubca1\ud130 \\(v\\in V\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(m,\\) \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[T\\left( \\frac{n}{m} v \\right) = \\frac{n}{m} T(v) \\tag{4}\\]
    \n\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 3\ub2e8\uacc4\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec
    \n\\[T(v) = T\\left( \\frac{m}{m} v \\right) = m T\\left( \\frac{1}{m} v \\right)\\]
    \n\uc774\ubbc0\ub85c
    \n\\[T\\left( \\frac{1}{m} v \\right) = \\frac{1}{m} T(v)\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub2e4\uc2dc 3\ub2e8\uacc4\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec
    \n\\[T\\left( \\frac{n}{m} v \\right) = nT \\left( \\frac{1}{m}v \\right) = \\frac{n}{m} T(v)\\]
    \n\uc774\ubbc0\ub85c (4)\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n

    5\ub2e8\uacc4.<\/span>
    \n\\(k\\in \\mathbb{Q},\\) \\(v\\in V\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(k=0\\)\uc774\uba74 1\ub2e8\uacc4\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec
    \n\\[T(kv) = kT(v) \\tag{5}\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \\(k > 0\\)\uc774\uba74 4\ub2e8\uacc4\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec (5)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \\(k < 0\\)\uc774\uba74 2\ub2e8\uacc4\uc640 4\ub2e8\uacc4\uc758 \uacb0\uacfc\ub97c \uc870\ud569\ud558\uc5ec (5)\ub97c \uc5bb\ub294\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \uc774\uc81c \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud574\ubcf4\uae30 \ubc14\ub780\ub2e4.<\/p>\n

      \n
    • \\(F\\)\uac00 \uc720\ud55c\uccb4(finite field)\uc77c \ub54c \uc704 \uc815\ub9ac\uac00 \uc131\ub9bd\ud560\uae4c?<\/li>\n
    • \\(F\\)\uac00 \uac00\uc0b0\uccb4(countable field)\uc77c \ub54c \uc704 \uc815\ub9ac\uac00 \uc131\ub9bd\ud560\uae4c?<\/li>\n<\/ul>\n

      \uc131\ub9bd\ud55c\ub2e4\uba74 \uc99d\uba85\ud574 \ubcf4\uc790. \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \ubc18\ub840\ub97c \ub4e4\uace0, \uc815\ub9ac\uac00 \uc131\ub9bd\ud558\ub3c4\ub85d \ud558\ub294 \\(F\\)\uc758 \uc870\uac74\uc744 \uad6c\ud574 \ubcf4\uc790.<\/p>\n","protected":false},"excerpt":{"rendered":"

      \\(V\\)\uc640 \\(W\\)\uac00 \uccb4 \\(F\\) \uc704\uc5d0\uc11c\uc758 \ubca1\ud130\uacf5\uac04\uc774\ub77c \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(T : V \\rightarrow W\\)\uac00 \ub450 \uc870\uac74 \uc784\uc758\uc758 \\(v_1 ,\\, v_2 \\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(T(v_1 + v_2 ) = T(v_1 ) + T(v_2 )\\)\uc774\ub2e4, \uc784\uc758\uc758 \\(k \\in F\\)\uc640 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(T(kv) = kT(v)\\)\uc774\ub2e4 \ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(T\\)\ub97c \uc120\ud615\ubcc0\ud658(linear transformation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(F = \\mathbb{R}\\)\ub77c\uba74, \\(T\\)\uac00 (1)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub354\ub77c\ub3c4 (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\uc744 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \\(F=\\mathbb{Q}\\)\ub77c\uba74…<\/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":[432,429,428,433,431,430],"class_list":["post-5401","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-432","tag-429","tag-428","tag-433","tag-431","tag-430"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5401","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=5401"}],"version-history":[{"count":20,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5401\/revisions"}],"predecessor-version":[{"id":5421,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5401\/revisions\/5421"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5401"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}