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1-12-2.html
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<article>
<h2 id="1-12-2">1-12-2 矩陣求解</h2>
<h3>Row Reduction</h3>
<h4>範例1</h4>
<img src="./assets/image/1-12-2/1.jpg" alt="" />
<h4>範例2</h4>
<img src="./assets/image/1-12-2/2.jpg" alt="" />
<h4>範例3</h4>
<img src="./assets/image/1-12-2/3.jpg" alt="" />
<h3>Row Echelon Form</h3>
<p>
線性代數中，一個矩陣如果符合下列條件的話，我們稱之為列階梯形矩陣或列梯形式矩陣（英語：Row
Echelon Form）：
</p>
<ul>
<li> 若某列有個非零元素，則必在任何全零列之上。</li>
<li>
某列最左邊的（即第一個）非零元素稱為首項係數（leading entry of
row）。某列的首項係數必定比上一列的首項係數更靠右。
</li>
<li>
因為首項係數要不是最靠右的，要不就是左邊都是零，所以根據上面二點，在首項係數所在的行中，在首項係數下面的元素都會是零。
</li>
</ul>
<img src="./assets/image/1-12-2/4.jpeg" alt="" />
<h3>Reduced Echelon Form</h3>
<p>
簡化行梯形形式（Reduced Row Echelon Form,
RREF）是一種進一步簡化的矩陣形式，它比行梯形形式（Row Echelon Form,
REF）有更嚴格的要求。矩陣處於簡化行梯形形式時，使解線性方程組變得更加直接和清晰。
</p>
<ul>
<li>除包含 Row Echelon Form 條件外，還會要求非零列的首項係數必須是 1[1]</li>
<li>每個 row 的首項係數縱向的其他成員需要為 0。</li>
</ul>
<img src="./assets/image/1-12-2/5.jpeg" alt="" />
<p>得到 Reduced Row Echelon Form, RREF 的2種方式：</p>
<ul>
<li>
<a href="https://www.geogebra.org/classic" target="_blank" rel="noopener noreferrer">
Geogebra-classic
</a>
</li>
<li>Algorithm</li>
</ul>
<h4>Geogebra-classic</h4>
<h4>步驟1</h4>
<img src="./assets/image/1-12-2/6.jpg" alt="" />
<h4>步驟2</h4>
<img src="./assets/image/1-12-2/7.jpg" alt="" />
<h4>步驟3</h4>
<img src="./assets/image/1-12-2/8.jpg" alt="" />
<h4>步驟4</h4>
<img src="./assets/image/1-12-2/9.jpg" alt="" />
<h4>步驟5</h4>
<img src="./assets/image/1-12-2/10.jpg" alt="" />
<h4>步驟6</h4>
<img src="./assets/image/1-12-2/11.jpg" alt="" />
<h4>Algorithm for Finding RREF</h4>
<ol>
<li>Change the leading entry of the next working row to 1.</li>
<li>Eliminate all numbers below the leading entry.</li>
<li>Keep repeating steps 1 and 2 until we find the leading entry for each row.</li>
<li>For each leading entry, eliminate all numbers above it.</li>
</ol>
<p>Example:</p>
<p>First, follow steps 1, 2, and 3.</p>
<img src="./assets/image/1-12-2/12.jpeg" alt="" />
<img src="./assets/image/1-12-2/13.jpg" alt="" />
<img src="./assets/image/1-12-2/14.jpg" alt="" />
<img src="./assets/image/1-12-2/15.jpeg" alt="" />
<p>Then, follow steps 4.</p>
<img src="./assets/image/1-12-2/16.jpeg" alt="" />
<img src="./assets/image/1-12-2/17.jpg" alt="" />
<img src="./assets/image/1-12-2/18.jpg" alt="" />
<h3>Reduced Row Echelon Form 的好處</h3>
<h4>第1個好處</h4>
<img src="./assets/image/1-12-2/19.jpg" alt="" />
<p>快速求解。</p>
<h4>第2個好處</h4>
<p>可以描述多解的組合。</p>
<p>舉例以下例子具有無限多個解，我們需要根據例子界定無限多解的範圍及特性。</p>
<img src="./assets/image/1-12-2/20.jpg" alt="" />
<img src="./assets/image/1-12-2/21.jpg" alt="" />
<img src="./assets/image/1-12-2/22.jpg" alt="" />
<img src="./assets/image/1-12-2/23.jpg" alt="" />
<img src="./assets/image/1-12-2/24.jpg" alt="" />
<p>
解完範例 rref 之後會發現該線性方程組矩陣 X3 有無限多種解。X3 代回第一個線性方程式來看， X1
也會根據 X3 的解的不同而變化。
</p>
<h5>名詞解釋:</h5>
<ul>
<li><strong>Pivots:In RREF,the pivots are the leading entry on each row.</strong></li>
<li><strong>Basic Variable: the variable on pivots position.</strong></li>
<li><strong>Free Variable: Not Basic Variable.</strong></li>
</ul>
<p></p>
</article>
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