import{_ as p,c as i,b as e,a as s,e as n,d as l,w as m,r,o as c}from"./app-CZkEq2xj.js";const o={};function h(g,a){const t=r("RouteLink");return c(),i("div",null,[a[7]||(a[7]=e('<h1 id="数学" tabindex="-1">数学</h1><h2 id="線形代数" tabindex="-1">線形代数</h2><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 実行列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A \\in \\mathbb{R}^{n \\times m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span></span> の第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> 列 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">j = 1, 2, \\dots, m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span></span></span></span>) を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_j \\in \\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> とする。<br> 各部分集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo>⊆</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">J \\subseteq \\{1, 2, \\dots, m\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">}</span></span></span></span> について、その要素数を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>J</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|J|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mord">∣</span></span></span></span> で表し、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>j</mi><mo>∈</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_j \\ (j \\in J)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mclose">)</span></span></span></span> を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> に関する昇順で左から並べて得られる <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> の部分行列を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi mathvariant="normal">∣</mi><mi>J</mi><mi mathvariant="normal">∣</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A[J] \\in \\mathbb{R}^{n \\times |J|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.0962em;">J</span><span class="mord mtight">∣</span></span></span></span></span></span></span></span></span></span></span></span> で表す。<br> 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のとき、"),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r"),s("mi",null,"a"),s("mi",null,"n"),s("mi",null,"k"),s("mo",{stretchy:"false"},"("),s("mi",null,"A"),s("mo",{stretchy:"false"},"["),s("mi",null,"I"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"∣")]),s("annotation",{encoding:"application/x-tex"},"rank(A[I]) = |I|")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0278em"}},"r"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0315em"}},"ank"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mopen"},"["),s("span",{class:"mord 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mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 上の関数</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mo>−</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f(x, y) = (x + y) \\exp\\left(-\\frac{x^2 + y^2}{2}\\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.4411em;vertical-align:-0.95em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>について次の各問いに答えよ。</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> の停留点を全て求めよ。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> の極大点と極小点を全て求めよ。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> の最大値または最小値が存在する場合、それらを求めよ。</li></ol><h2 id="微分方程式" tabindex="-1">微分方程式</h2><p>次の微分方程式の一般解を求めよ。</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>x</mi><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} = \\frac{x^2 + 6y^2}{4xy} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">6</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} = e^{2x + y + 1} - 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span 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u=p(o,[["render",h]]),y=JSON.parse('{"path":"/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03ist.html","title":"数学","lang":"zh-CN","frontmatter":{"tags":["過去問","例题","数学"],"dlink":["[[R03ist.pdf]]"],"datetime":"2024-12-10T01:22:51.000Z","author":["Cyletix"],"description":"数学 線形代数 n×m 実行列 A∈Rn×m の第 j 列 (j=1,2,…,m) を aj​∈Rn とする。 各部分集合 J⊆{1,2,…,m} について、その要素数を ∣J∣ で表し、aj​ (j∈J) を j に関する昇順で左から並べて得られる A の部分行列を A[J]∈Rn×∣J∣ で表す。 このとき、以下の問いに答えよ。 以下の行列 A に...","head":[["script",{"type":"application/ld+json"},"{\\"@context\\":\\"https://schema.org\\",\\"@type\\":\\"Article\\",\\"headline\\":\\"数学\\",\\"image\\":[\\"\\"],\\"dateModified\\":null,\\"author\\":[{\\"@type\\":\\"Person\\",\\"name\\":\\"Cyletix\\"}]}"],["meta",{"property":"og:url","content":"https://LincDocs.github.io/Pkmer-Math/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03ist.html"}],["meta",{"property":"og:site_name","content":"Pkmer-Math"}],["meta",{"property":"og:title","content":"数学"}],["meta",{"property":"og:description","content":"数学 線形代数 n×m 実行列 A∈Rn×m の第 j 列 (j=1,2,…,m) を aj​∈Rn とする。 各部分集合 J⊆{1,2,…,m} について、その要素数を ∣J∣ で表し、aj​ (j∈J) を j に関する昇順で左から並べて得られる A の部分行列を A[J]∈Rn×∣J∣ で表す。 このとき、以下の問いに答えよ。 以下の行列 A に..."}],["meta",{"property":"og:type","content":"article"}],["meta",{"property":"og:locale","content":"zh-CN"}],["meta",{"property":"article:author","content":"Cyletix"}],["meta",{"property":"article:tag","content":"数学"}],["meta",{"property":"article:tag","content":"例题"}],["meta",{"property":"article:tag","content":"過去問"}]],"tag":["過去問","例题","数学"]},"git":{},"readingTime":{"minutes":1.27,"words":381},"filePathRelative":"Other/例题/九州過去問/R03ist.md","excerpt":"\\n<h2>線形代数</h2>\\n<p><span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding=\\"application/x-tex\\">n \\\\times m</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.6667em;vertical-align:-0.0833em;\\"></span><span class=\\"mord mathnormal\\">n</span><span class=\\"mspace\\" style=\\"margin-right:0.2222em;\\"></span><span class=\\"mbin\\">×</span><span class=\\"mspace\\" style=\\"margin-right:0.2222em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.4306em;\\"></span><span class=\\"mord mathnormal\\">m</span></span></span></span> 実行列 <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant=\\"double-struck\\">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></mrow><annotation encoding=\\"application/x-tex\\">A \\\\in \\\\mathbb{R}^{n \\\\times m}</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.7224em;vertical-align:-0.0391em;\\"></span><span class=\\"mord mathnormal\\">A</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">∈</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.7713em;\\"></span><span class=\\"mord\\"><span class=\\"mord mathbb\\">R</span><span class=\\"msupsub\\"><span class=\\"vlist-t\\"><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.7713em;\\"><span style=\\"top:-3.063em;margin-right:0.05em;\\"><span class=\\"pstrut\\" style=\\"height:2.7em;\\"></span><span class=\\"sizing reset-size6 size3 mtight\\"><span class=\\"mord mtight\\"><span class=\\"mord mathnormal mtight\\">n</span><span class=\\"mbin mtight\\">×</span><span class=\\"mord mathnormal mtight\\">m</span></span></span></span></span></span></span></span></span></span></span></span> の第 <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>j</mi></mrow><annotation encoding=\\"application/x-tex\\">j</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.854em;vertical-align:-0.1944em;\\"></span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0572em;\\">j</span></span></span></span> 列 (<span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator=\\"true\\">,</mo><mn>2</mn><mo separator=\\"true\\">,</mo><mo>…</mo><mo separator=\\"true\\">,</mo><mi>m</mi></mrow><annotation encoding=\\"application/x-tex\\">j = 1, 2, \\\\dots, m</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.854em;vertical-align:-0.1944em;\\"></span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0572em;\\">j</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">=</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.8389em;vertical-align:-0.1944em;\\"></span><span class=\\"mord\\">1</span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mord\\">2</span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"minner\\">…</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mord mathnormal\\">m</span></span></span></span>) を <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>∈</mo><msup><mi mathvariant=\\"double-struck\\">R</mi><mi>n</mi></msup></mrow><annotation encoding=\\"application/x-tex\\">a_j \\\\in \\\\mathbb{R}^n</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.8252em;vertical-align:-0.2861em;\\"></span><span class=\\"mord\\"><span class=\\"mord mathnormal\\">a</span><span class=\\"msupsub\\"><span class=\\"vlist-t vlist-t2\\"><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.3117em;\\"><span style=\\"top:-2.55em;margin-left:0em;margin-right:0.05em;\\"><span class=\\"pstrut\\" style=\\"height:2.7em;\\"></span><span class=\\"sizing reset-size6 size3 mtight\\"><span class=\\"mord mathnormal mtight\\" style=\\"margin-right:0.0572em;\\">j</span></span></span></span><span class=\\"vlist-s\\">​</span></span><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.2861em;\\"><span></span></span></span></span></span></span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">∈</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.6889em;\\"></span><span class=\\"mord\\"><span class=\\"mord mathbb\\">R</span><span class=\\"msupsub\\"><span class=\\"vlist-t\\"><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.6644em;\\"><span style=\\"top:-3.063em;margin-right:0.05em;\\"><span class=\\"pstrut\\" style=\\"height:2.7em;\\"></span><span class=\\"sizing reset-size6 size3 mtight\\"><span class=\\"mord mathnormal mtight\\">n</span></span></span></span></span></span></span></span></span></span></span> とする。<br>\\n各部分集合 <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>J</mi><mo>⊆</mo><mo stretchy=\\"false\\">{</mo><mn>1</mn><mo separator=\\"true\\">,</mo><mn>2</mn><mo separator=\\"true\\">,</mo><mo>…</mo><mo separator=\\"true\\">,</mo><mi>m</mi><mo stretchy=\\"false\\">}</mo></mrow><annotation encoding=\\"application/x-tex\\">J \\\\subseteq \\\\{1, 2, \\\\dots, m\\\\}</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.8193em;vertical-align:-0.136em;\\"></span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0962em;\\">J</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">⊆</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:1em;vertical-align:-0.25em;\\"></span><span class=\\"mopen\\">{</span><span class=\\"mord\\">1</span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mord\\">2</span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"minner\\">…</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mpunct\\">,</span><span class=\\"mspace\\" style=\\"margin-right:0.1667em;\\"></span><span class=\\"mord mathnormal\\">m</span><span class=\\"mclose\\">}</span></span></span></span> について、その要素数を <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi mathvariant=\\"normal\\">∣</mi><mi>J</mi><mi mathvariant=\\"normal\\">∣</mi></mrow><annotation encoding=\\"application/x-tex\\">|J|</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:1em;vertical-align:-0.25em;\\"></span><span class=\\"mord\\">∣</span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0962em;\\">J</span><span class=\\"mord\\">∣</span></span></span></span> で表し、<span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy=\\"false\\">(</mo><mi>j</mi><mo>∈</mo><mi>J</mi><mo stretchy=\\"false\\">)</mo></mrow><annotation encoding=\\"application/x-tex\\">a_j \\\\ (j \\\\in J)</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:1.0361em;vertical-align:-0.2861em;\\"></span><span class=\\"mord\\"><span class=\\"mord mathnormal\\">a</span><span class=\\"msupsub\\"><span class=\\"vlist-t vlist-t2\\"><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.3117em;\\"><span style=\\"top:-2.55em;margin-left:0em;margin-right:0.05em;\\"><span class=\\"pstrut\\" style=\\"height:2.7em;\\"></span><span class=\\"sizing reset-size6 size3 mtight\\"><span class=\\"mord mathnormal mtight\\" style=\\"margin-right:0.0572em;\\">j</span></span></span></span><span class=\\"vlist-s\\">​</span></span><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.2861em;\\"><span></span></span></span></span></span></span><span class=\\"mspace\\"> </span><span class=\\"mopen\\">(</span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0572em;\\">j</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">∈</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:1em;vertical-align:-0.25em;\\"></span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0962em;\\">J</span><span class=\\"mclose\\">)</span></span></span></span> を <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>j</mi></mrow><annotation encoding=\\"application/x-tex\\">j</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.854em;vertical-align:-0.1944em;\\"></span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0572em;\\">j</span></span></span></span> に関する昇順で左から並べて得られる <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>A</mi></mrow><annotation encoding=\\"application/x-tex\\">A</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.6833em;\\"></span><span class=\\"mord mathnormal\\">A</span></span></span></span> の部分行列を <span v-pre class=\\"katex\\"><span class=\\"katex-mathml\\"><math xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><semantics><mrow><mi>A</mi><mo stretchy=\\"false\\">[</mo><mi>J</mi><mo stretchy=\\"false\\">]</mo><mo>∈</mo><msup><mi mathvariant=\\"double-struck\\">R</mi><mrow><mi>n</mi><mo>×</mo><mi mathvariant=\\"normal\\">∣</mi><mi>J</mi><mi mathvariant=\\"normal\\">∣</mi></mrow></msup></mrow><annotation encoding=\\"application/x-tex\\">A[J] \\\\in \\\\mathbb{R}^{n \\\\times |J|}</annotation></semantics></math></span><span class=\\"katex-html\\" aria-hidden=\\"true\\"><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:1em;vertical-align:-0.25em;\\"></span><span class=\\"mord mathnormal\\">A</span><span class=\\"mopen\\">[</span><span class=\\"mord mathnormal\\" style=\\"margin-right:0.0962em;\\">J</span><span class=\\"mclose\\">]</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span><span class=\\"mrel\\">∈</span><span class=\\"mspace\\" style=\\"margin-right:0.2778em;\\"></span></span><span class=\\"base\\"><span class=\\"strut\\" style=\\"height:0.888em;\\"></span><span class=\\"mord\\"><span class=\\"mord mathbb\\">R</span><span class=\\"msupsub\\"><span class=\\"vlist-t\\"><span class=\\"vlist-r\\"><span class=\\"vlist\\" style=\\"height:0.888em;\\"><span style=\\"top:-3.063em;margin-right:0.05em;\\"><span class=\\"pstrut\\" style=\\"height:2.7em;\\"></span><span class=\\"sizing reset-size6 size3 mtight\\"><span class=\\"mord mtight\\"><span class=\\"mord mathnormal mtight\\">n</span><span class=\\"mbin mtight\\">×</span><span class=\\"mord mtight\\">∣</span><span class=\\"mord mathnormal mtight\\" style=\\"margin-right:0.0962em;\\">J</span><span class=\\"mord mtight\\">∣</span></span></span></span></span></span></span></span></span></span></span></span> で表す。<br>\\nこのとき、以下の問いに答えよ。</p>","autoDesc":true,"bioChainData":{"outlink":[{"title":"R03-数学-向量分析","link":"Other/例题/九州過去問/R03-数学-向量分析.html"},{"title":"R03-数学-复素数","link":"Other/例题/九州過去問/R03-数学-复素数.html"},{"title":"R03-数学-概率统计","link":"Other/例题/九州過去問/R03-数学-概率统计.html"},{"title":"R03-B-問1","link":"Other/例题/九州過去問/R03-B-問1.html"},{"title":"R03-B-問2","link":"Other/例题/九州過去問/R03-B-問2.html"},{"title":"R03-C-問1","link":"Other/例题/九州過去問/R03-C-問1.html"},{"title":"R03-C-問2","link":"Other/例题/九州過去問/R03-C-問2.html"}],"backlink":[],"localMap":{"nodes":[{"id":"Other/例题/九州過去問/R03ist.md","value":{"title":"R03ist","path":"Other/例题/九州過去問/R03ist.md","outlink":["Other/例题/九州過去問/R03-数学-向量分析.md","Other/例题/九州過去問/R03-数学-复素数.md","Other/例题/九州過去問/R03-数学-概率统计.md","Other/例题/九州過去問/R03-B-問1.md","Other/例题/九州過去問/R03-B-問2.md","Other/例题/九州過去問/R03-C-問1.md","Other/例题/九州過去問/R03-C-問2.md"],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-数学-向量分析.md","value":{"title":"R03-数学-向量分析","path":"Other/例题/九州過去問/R03-数学-向量分析.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-数学-复素数.md","value":{"title":"R03-数学-复素数","path":"Other/例题/九州過去問/R03-数学-复素数.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-数学-概率统计.md","value":{"title":"R03-数学-概率统计","path":"Other/例题/九州過去問/R03-数学-概率统计.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-B-問1.md","value":{"title":"R03-B-問1","path":"Other/例题/九州過去問/R03-B-問1.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-B-問2.md","value":{"title":"R03-B-問2","path":"Other/例题/九州過去問/R03-B-問2.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-C-問1.md","value":{"title":"R03-C-問1","path":"Other/例题/九州過去問/R03-C-問1.md","outlink":[],"backlink":[]}},{"id":"Other/例题/九州過去問/R03-C-問2.md","value":{"title":"R03-C-問2","path":"Other/例题/九州過去問/R03-C-問2.md","outlink":[],"backlink":[]}}],"links":[{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-数学-向量分析.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-数学-复素数.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-数学-概率统计.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-B-問1.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-B-問2.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-C-問1.md"},{"source":"Other/例题/九州過去問/R03ist.md","target":"Other/例题/九州過去問/R03-C-問2.md"}]}}}');export{u 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