https://yukar.icu/column/undergraduate/Recent content in Undergraduate on Yukari's BlogHugozh-cnSun, 12 Jul 2026 00:00:00 +0000https://yukar.icu/posts/linearalgebra/1/Sun, 12 Jul 2026 00:00:00 +0000https://yukar.icu/posts/linearalgebra/1/<h2 id="matrix">Matrix</h2> <h3 id="elementary-matrix">Elementary Matrix</h3> <p>The standard basis of $\mathbb F^n$ is $\{e_i\}_{1\leq i\leq n}$, where $e_i$ is the vector with $1$ in the $i$-th entry and $0$ elsewhere. It holds </p> $$e_i^\top Ae_j=a_{ij},$$<p> This can be a test of a linear map $A:\mathbb F^n\to \mathbb F^n$ by the standard basis of $\mathbb F^n$. In total, </p> $$A=\sum_{i,j=1}^n a_{ij}e_ie^\top_j=\sum_{i,j=1}^n a_{ij}E_{ij},$$<p>We introduce a kind of basis of $M_n(\mathbb F)=\mathbb F^{n\times n}$ as $\{E_{ij}\}_{1\leq i,j\leq n}$, where $E_{ij}$ is the matrix with $1$ in the $(i,j)$-th entry and $0$ elsewhere, that is so-called elementary matrix, satisfying </p>