## 2016年7月7日木曜日

### 数学 - 線型代数 - 線型写像 - 線型写像の定義と例(線型かどうか)

• 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

1. $\begin{array}{l}F\left({x}_{1}+{x}_{2},{y}_{1}+{y}_{2}\right)\\ =\left({x}_{1}+{x}_{2}+1,{y}_{1}+{y}_{2}\right)\\ =\left({x}_{1}+1,{y}_{1}\right)+\left({x}_{2}+1,{y}_{2}\right)+\left(-1,0\right)\\ =F\left({x}_{1},{y}_{1}\right)+F\left({x}_{2},{y}_{2}\right)+\left(-1,0\right)\\ 線型ではない。\end{array}$

2. $\begin{array}{l}F\left({x}_{1}+{x}_{2},{y}_{1}+{y}_{2}\right)\\ =\left({y}_{1}+{y}_{2},{x}_{1}+{x}_{2}\right)\\ =\left({y}_{1},{x}_{1}\right)+\left({y}_{2},{x}_{2}\right)\\ =F\left({x}_{1},{y}_{1}\right)+F\left({x}_{2},{y}_{2}\right)\\ 線型。\end{array}$

3. $\begin{array}{l}F\left({x}_{1}+{x}_{2},{y}_{1}+{y}_{2}\right)\\ =\left({x}_{1}+{x}_{2},-2\left({y}_{1}+{y}_{2}\right),0\right)\\ =\left({x}_{1},2{y}_{1},0\right)+\left({x}_{2},-2{y}_{2},0\right)\\ =F\left({x}_{1},{y}_{1}\right)+F\left({x}_{2},{y}_{2}\right)\\ 線型。\end{array}$

4. $\begin{array}{l}F\left({x}_{1}+{x}_{2},{y}_{1}+{y}_{2}\right)\\ =\left({\left({x}_{1}+{x}_{2}\right)}^{2},{y}_{1}+{y}_{2}\right)\\ =\left({x}_{1}^{2},{y}_{1}\right)+\left({x}_{2}{}^{2},{y}_{2}\right)+\left(2{x}_{1}{x}_{2},0\right)\\ =F\left({x}_{1},{y}_{1}\right)+F\left({x}_{2},{y}_{2}\right)+\left(2{x}_{1}{x}_{2},0\right)\\ 線型ではない。\end{array}$

5. $\begin{array}{l}F\left({x}_{1}+{x}_{2},{y}_{1}+{y}_{2},{z}_{1}+{z}_{2}\right)\\ =\left({y}_{1}+{y}_{2}+{z}_{1}+{z}_{2},{z}_{1}+{z}_{2}+{x}_{1}+{x}_{2}\right)\\ =\left({y}_{1}+{z}_{1},{z}_{1}+{x}_{1}\right)+\left({y}_{2}+{z}_{2},{z}_{2}+{x}_{2}\right)\\ =F\left({x}_{1},{y}_{1},{z}_{1}\right)+F\left({x}_{2},{y}_{2},{z}_{2}\right)\\ 線型。\end{array}$

6. $\begin{array}{l}F\left({x}_{1}+{x}_{2}\right)\\ =a\left({x}_{1}+{x}_{2}\right)+b\\ =a{x}_{1}+a{x}_{2}+b+b-b\\ =F\left({x}_{1}\right)+F\left({x}_{2}\right)-b\\ b=0のとき、線型。\\ b\ne 0のとき、線型ではない。\end{array}$