https://www.chenk.top/en/tags/orthogonality/Recent content in Orthogonality on Chen Kai BlogHugoenWed, 12 Feb 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/07-orthogonality-and-projections/Wed, 12 Feb 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/07-orthogonality-and-projections/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/linear-algebra/07-orthogonality-and-projections/illustration_1.png" alt="Essence of Linear Algebra (7): Orthogonality and Projections — When Vectors Mind Their Own Business — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="why-orthogonality-matters" class="heading-anchor">Why Orthogonality Matters<a href="#why-orthogonality-matters" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>Two vectors are <strong>orthogonal</strong> when they &ldquo;do not interfere&rdquo; with one another. That single idea — one direction tells you nothing about the other — powers GPS positioning, noise-canceling headphones, JPEG compression, recommendation systems, and most of numerical linear algebra.</p> <p>Orthogonality is the single biggest computational shortcut in linear algebra. With a generic basis, finding coordinates is solving a linear system. With an <strong>orthogonal</strong> basis, finding coordinates is one dot product per axis. Hard problem, easy problem, same problem — just a better basis.</p>