article ja sentence
  cons anno text "ベクトル" cons text "vector" nil
  cons text "の"
  cons anno ruby text "線型結合" text "せんけいけつごう" cons text "linear combination" nil
  cons text "と"
  cons anno text "スパン" cons text "span" nil
  nil
body
  cons section linear_combination sentence
    cons anno ruby text "線型結合" text "せんけいけつごう" cons text "linear combination" nil
    nil
    body
      cons paragraph
        cons parallel
          cons variant ja "いくつかの{ベクトル/vector}をそれぞれ{[定数倍/ていすうばい]/scalar multiplication}して[足/た]し[合/あ]わせることを、{[線型結合/せんけいけつごう]/linear combination}といいます。"
          cons variant en "A linear combination is formed by multiplying each of several vectors by a scalar and adding the results."
          nil
        cons parallel
          cons variant ja sentence
            cons text "それぞれの"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "に"
            cons ruby text "掛" text "か"
            cons text "ける"
            cons ruby text "数" text "かず"
            cons text "が、"
            cons anno ruby text "係数" text "けいすう" cons text "coefficient" nil
            cons text "です。"
            nil
          cons variant en sentence
            cons text "The number multiplying each vector is called its coefficient."
            nil
          nil
        nil
    nil
  cons section span sentence
    cons anno text "スパン" cons text "span" nil
    nil
    body
      cons paragraph
        cons parallel
          cons variant ja sentence
            cons anno ruby text "係数" text "けいすう" cons text "coefficient" nil
            cons text "を"
            cons ruby text "自由" text "じゆう"
            cons text "に"
            cons ruby text "変" text "か"
            cons text "えて、"
            cons ref linear_combination anno ruby text "線型結合" text "せんけいけつごう" cons text "linear combination" nil
            cons text "で"
            cons ruby text "作" text "つく"
            cons text "れる"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "を"
            cons strong text "すべて"
            cons ruby text "集" text "あつ"
            cons text "めたものが、"
            cons ruby text "元" text "もと"
            cons text "の"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "たちの"
            cons anno text "スパン" cons text "span" nil
            cons text "です。"
            nil
          cons variant en sentence
            cons text "The span of a collection of vectors is the set of "
            cons strong text "all vectors"
            cons text " that can be formed as "
            cons ref linear_combination text "linear combinations"
            cons text " of them, allowing the coefficients to vary freely."
            nil
          nil
        cons parallel
          cons variant ja sentence
            cons anno ruby text "係数" text "けいすう" cons text "coefficient" nil
            cons text "を"
            cons ruby text "一組" text "ひとくみ"
            cons text "だけ"
            cons ruby text "決" text "き"
            cons text "めて"
            cons ruby text "得" text "え"
            cons text "られる"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "と、そのすべての"
            cons ruby text "可能性" text "かのうせい"
            cons text "を"
            cons ruby text "集" text "あつ"
            cons text "めた"
            cons anno text "スパン" cons text "span" nil
            cons text "を"
            cons ruby text "区別" text "くべつ"
            cons text "しましょう。"
            nil
          cons variant en sentence
            cons text "Distinguish a single vector obtained from one choice of coefficients from the span, which contains all possible results."
            nil
          nil
        nil
    nil
  cons section plane sentence
    cons anno concat
      cons ruby text "平面" text "へいめん"
      cons text "での"
      cons ruby text "例" text "れい"
      nil
      cons text "examples in the plane" nil
    nil
    body
      cons paragraph
        cons parallel
          cons variant ja sentence
            cons text "たとえば"
            cons anno ruby text "平面" text "へいめん" cons text "plane" nil
            cons text "では、0でない"
            cons ruby text "一本" text "いっぽん"
            cons text "の"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "の"
            cons anno text "スパン" cons text "span" nil
            cons text "は、"
            cons anno ruby text "原点" text "げんてん" cons text "origin" nil
            cons text "を"
            cons ruby text "通" text "とお"
            cons text "る"
            cons ruby text "一本" text "いっぽん"
            cons text "の"
            cons anno ruby text "直線" text "ちょくせん" cons text "line" nil
            cons text "です。"
            nil
          cons variant en sentence
            cons text "For example, in the plane, the span of one nonzero vector is a line through the origin."
            nil
          nil
        cons parallel
          cons variant ja sentence
            cons ruby text "同" text "おな"
            cons text "じ"
            cons ruby text "方向" text "ほうこう"
            cons text "や"
            cons ruby text "逆方向" text "ぎゃくほうこう"
            cons text "の"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "を"
            cons ruby text "何本" text "なんぼん"
            cons ruby text "加" text "くわ"
            cons text "えても、この"
            cons anno text "スパン" cons text "span" nil
            cons text "は"
            cons ruby text "変" text "か"
            cons text "わりません。"
            nil
          cons variant en sentence
            cons text "Adding any number of vectors in the same or opposite direction leaves this span unchanged."
            nil
          nil
        cons parallel
          cons variant ja sentence
            cons anno ruby text "平行" text "へいこう" cons text "parallel" nil
            cons text "でない"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "を"
            cons ruby text "加" text "くわ"
            cons text "えると、"
            cons anno text "スパン" cons text "span" nil
            cons text "は"
            cons anno ruby text "平面" text "へいめん" cons text "plane" nil
            cons ruby text "全体" text "ぜんたい"
            cons text "になります。"
            nil
          cons variant en sentence
            cons text "Adding a vector that is not parallel to that line makes the span the entire plane."
            nil
          nil
        nil
      cons table cons left cons left cons right nil
        some row
          cons sentence
            cons anno concat
              cons ruby text "平面内" text "へいめんない"
              cons text "のベクトル"
              nil
              cons text "vectors in the plane" nil
            nil
          cons sentence
            cons anno text "スパン" cons text "span" nil
            nil
          cons sentence
            cons anno ruby text "次元" text "じげん" cons text "dimension" nil
            nil
          nil
        cons row
          cons sentence
            cons anno concat
              cons text "0でない"
              cons ruby text "一本" text "いっぽん"
              nil
              cons text "one nonzero vector" nil
            nil
          cons sentence
            cons anno concat
              cons ruby text "原点" text "げんてん"
              cons text "を"
              cons ruby text "通" text "とお"
              cons text "る"
              cons ruby text "直線" text "ちょくせん"
              nil
              cons text "a line through the origin" nil
            nil
          cons sentence cons text "1" nil
          nil
        cons row
          cons sentence
            cons anno concat
              cons ruby text "平行" text "へいこう"
              cons text "な"
              cons ruby text "二本" text "にほん"
              cons text "（"
              cons ruby text "少" text "すく"
              cons text "なくとも"
              cons ruby text "一本" text "いっぽん"
              cons text "は0でない）"
              nil
              cons text "two parallel vectors, at least one nonzero" nil
            nil
          cons sentence
            cons anno concat
              cons ruby text "原点" text "げんてん"
              cons text "を"
              cons ruby text "通" text "とお"
              cons text "る"
              cons ruby text "直線" text "ちょくせん"
              nil
              cons text "a line through the origin" nil
            nil
          cons sentence cons text "1" nil
          nil
        cons row
          cons sentence
            cons anno concat
              cons ruby text "平行" text "へいこう"
              cons text "でない"
              cons ruby text "二本" text "にほん"
              nil
              cons text "two nonparallel vectors" nil
            nil
          cons sentence
            cons anno ruby text "平面全体" text "へいめんぜんたい" cons text "the entire plane" nil
            nil
          cons sentence cons text "2" nil
          nil
        nil
    nil
  cons section independence sentence
    cons anno concat
      cons ruby text "一次独立" text "いちじどくりつ"
      cons text "と"
      cons ruby text "次元" text "じげん"
      nil
      cons text "linear independence and dimension" nil
    nil
    body
      cons paragraph
        cons parallel
          cons variant ja sentence
            cons text "このように、"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "の"
            cons ruby text "本数" text "ほんすう"
            cons text "が"
            cons ruby text "増" text "ふ"
            cons text "えても、"
            cons anno text "スパン" cons text "span" nil
            cons text "の"
            cons anno ruby text "次元" text "じげん" cons text "dimension" nil
            cons text "が"
            cons ruby text "増" text "ふ"
            cons text "えるとは"
            cons ruby text "限" text "かぎ"
            cons text "りません。"
            nil
          cons variant en sentence
            cons text "Increasing the number of vectors does not necessarily increase the dimension of their span."
            nil
          nil
        cons parallel
          cons variant ja sentence
            cons anno text "スパン" cons text "span" nil
            cons text "の"
            cons anno ruby text "次元" text "じげん" cons text "dimension" nil
            cons text "は、"
            cons ruby text "元" text "もと"
            cons text "の"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "たちから"
            cons ruby text "選" text "えら"
            cons text "べる"
            cons anno ruby text "一次独立" text "いちじどくりつ" cons text "linearly independent" nil
            cons text "な"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "の"
            cons ruby text "最大本数" text "さいだいほんすう"
            cons text "で"
            cons ruby text "決" text "き"
            cons text "まります。"
            nil
          cons variant en sentence
            cons text "The dimension of the span is the largest number of linearly independent vectors that can be selected from the original collection."
            nil
          nil
        cons parallel
          cons variant ja sentence
            cons anno ruby text "一次独立" text "いちじどくりつ" cons text "linearly independent" nil
            cons text "とは、"
            cons ruby text "選" text "えら"
            cons text "んだどの"
            cons ruby text "一本" text "いっぽん"
            cons text "も、"
            cons ruby text "残" text "のこ"
            cons text "りの"
            cons anno text "ベクトル" cons text "vector" nil
            cons text "の"
            cons anno ruby text "線型結合" text "せんけいけつごう" cons text "linear combination" nil
            cons text "では"
            cons ruby text "作" text "つく"
            cons text "れないということです。"
            nil
          cons variant en sentence
            cons text "The selected vectors are linearly independent if none of them can be formed as a linear combination of the others."
            nil
          nil
        nil
    nil
  nil