https://www.chenk.top/en/tags/rank/Recent content in Rank on Chen Kai BlogHugoenWed, 29 Jan 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/05-linear-systems-and-column-space/Wed, 29 Jan 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/05-linear-systems-and-column-space/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/linear-algebra/05-linear-systems-and-column-space/illustration_1.png" alt="Essence of Linear Algebra (5): Linear Systems and Column Space — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="the-central-question" class="heading-anchor">The Central Question<a href="#the-central-question" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>Almost everything in applied mathematics eventually lands on the same question:</p> <blockquote> <p>Given a matrix <span class="math-inline">$A$</span> and a vector <span class="math-inline">$\vec{b}$</span> , does the equation <span class="math-inline">$A\vec{x} = \vec{b}$</span> have a solution? If so, how many?</p> </blockquote> <p>The mechanical answer is &ldquo;row-reduce and look.&rdquo; The <em>structural</em> answer is far more interesting — and it is the goal of this chapter. Three geometric objects tell you everything:</p>