## 2016年7月30日土曜日

### 数学 - 線型代数 - 線型写像 - 行列の積(一般化、数学的帰納法)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

$\begin{array}{l}A=\left(\begin{array}{cc}1& a\\ 0& 2\end{array}\right)\\ {A}^{2}=\left(\begin{array}{cc}1& 3a\\ 0& 4\end{array}\right)\\ {A}^{3}=\left(\begin{array}{cc}1& 7a\\ 0& 8\end{array}\right)\\ {A}^{4}=\left(\begin{array}{cc}1& 15a\\ 0& 16\end{array}\right)\\ {A}^{5}=\left(\begin{array}{cc}1& 31a\\ 0& 32\end{array}\right)\\ \\ {A}^{n}=\left(\begin{array}{cc}1& \left({2}^{n}-1\right)a\\ 0& {2}^{n}\end{array}\right)\\ \\ n=1\\ A=\left(\begin{array}{cc}1& a\\ 0& 2\end{array}\right)\\ n=k\\ {A}^{k}=\left(\begin{array}{cc}1& \left({2}^{k}-1\right)a\\ 0& {2}^{k}\end{array}\right)\\ {A}^{k+1}=\left(\begin{array}{cc}1& \left({2}^{k}-1\right)a\\ 0& {2}^{k}\end{array}\right)A\\ =\left(\begin{array}{cc}1& a+2\left({2}^{k}-1\right)a\\ 0& 2·{2}^{k}\end{array}\right)\\ =\left(\begin{array}{cc}1& {2}^{k+1}a-a\\ 0& {2}^{k+1}\end{array}\right)\\ =\left(\begin{array}{cc}1& \left({2}^{k+1}-1\right)a\\ 0& {2}^{k+1}\end{array}\right)\\ \\ {A}^{n}=\left(\begin{array}{cc}1& \left({2}^{n}-1\right)a\\ 0& {2}^{n}\end{array}\right)\left(n=1,2,···\right)\end{array}$