3.9 演習問題 〜 部分空間

問 3.40 (部分空間)   $\mathbb{R}^3$ の部分集合 $W_{1}$, $W_{2}$ が $\mathbb{R}^3$ の部分空間であるかどうか示せ．

(1)      $${ \mathbb{R}^3\supset W_{1}= \left\{ \left. \vec{x}= \begin{bmatr... ...a_{33} \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\vec{0} \right\}}$$

(2)      $$\displaystyle{ \mathbb{R}^3\supset W_{2}= \left\{ \left. \vec... ...y}{l} x + y - z\leq 1 \\ 2x -y + z \leq 0 \end{array}\right\}}$$

(3)      $$\displaystyle{ \mathbb{R}^3\supset W_{2}= \left\{ \left. \vec... ...}{l} x - y + z \leq 0 \\ x +2y -3z \leq 1 \end{array}\right\}}$$

問 3.41 (部分空間)   次の $\mathbb{R}^3$ の部分集合 $W_1$, $W_2$, $\cdots$, $W_5$ のうち 部分空間となるものはどれか．
    (1)   $$\displaystyle{ W_{1}= \left\{ \vec{x}\in\mathbb{R}^3 \left\ve... ...\! & & & \!-\! & \!x_{3}\! & \!=0 \end{array}\right. \right\} }$$
    (2)   $$\displaystyle{ W_{2}= \left\{ \vec{x}\in\mathbb{R}^3 \left\ve... ...! & & & \!+\! & \!3x_{3}\! & \!=3 \end{array}\right. \right\} }$$
    (3)   $$\displaystyle{ W_{3}= \left\{ \vec{x}\in\mathbb{R}^3 \left\ve... ...}\! & \!+\! & \!3x_{3}\! & \!\ge0 \end{array}\right. \right\} }$$
    (4)   $${ W_{4}= \left\{ \vec{x}\in\mathbb{R}^3 \left\vert\, x_{1}{}^2+ x_{2}{}^2+ x_{3}{}^2\leq1 \right. \right\} }$$
    (5)   $${ W_{5}= \left\{ \vec{x}\in\mathbb{R}^3 \left\vert\, \left( \begi... ...in{bmatrix} 1 \\ [-1ex] 1 \\ [-1ex] 1 \end{bmatrix}\right)=0 \right. \right\} }$$

問 3.42 (部分空間)   ベクトル空間 $\mathbb{R}^n$ において， 部分集合 $W$ は部分空間であるか述べよ．

(1) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{A\vec{x}=\vec{0}}\,\right\}}$$      (2) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{A\vec{x}=\vec{b}}\,\right\}}$$ （ただし， $\vec{b}\neq\vec{0}$）
(3) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{\Vert\vec{x}\Vert=1}\,\right\}}$$      (4) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{\Vert\vec{x}\Vert\le1}\,\right\}}$$
(5) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{(\vec{x},\vec{a})=0}\,\right\}}$$ （ $\vec{a}\in\mathbb{R}^{n}$ はあるベクトル）
(6) $${ W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{x_{1}\le1,x_{2}\le1,\cdots,x_{n}\le1}\,\right\}}$$

問 3.43 (部分空間)   ベクトル空間 $\mathbb{R}^2$ において， 部分集合 $W$ は部分空間であるか述べよ．

(1) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{2x_{1}-x_{2}=0}\,\right\}}$$         (2) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{2x_{1}-x_{2}=3}\,\right\}}$$

(3) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{x_{1}-x_{2}=0}\,\right\}}$$         (4) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{x_{1}+x_{2}=1}\,\right\}}$$

(5) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{x_{1}=0,x_{2}=0}\,\right\}}$$         (6) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{x_{1}>0,x_{2}>0}\,\right\}}$$

(7) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}... ...ay}{r} 2x_{1} + 3x_{2} \geq 0 \\ 4x_{1} + x_{2} \geq 0 \end{array}}\,\right\}}$$         (8) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}... ...ay}{r} 2x_{1} + 3x_{2} \geq 1 \\ 4x_{1} + x_{2} \geq 1 \end{array}}\,\right\}}$$

(9) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\in \mathbb{R}^2}\,\,\right\vert\,\,{x_{1}{}^2+x_{2}{}^2\leq 1}\,\right\}}$$     (10) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}... ...1} \\ x_{2} \end{bmatrix}\!=\! \begin{bmatrix} 0 \\ 0 \end{bmatrix}}\,\right\}}$$

(11) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}... ...{2} \end{bmatrix}\!=\! \begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix}}\,\right\}}$$

問 3.44 (部分空間)   ベクトル空間 $\mathbb{R}^3$ において， 部分集合 $W$ は部分空間であるか述べよ．

(1) $${ W=\left\{\left.\,{\begin{bmatrix}c_1 \\ c_2 \\ 0 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c_1,c_2\in\mathbb{R}}\,\right\}}$$         (2) $${ W=\left\{\left.\,{\begin{bmatrix}c \\ -c \\ 0 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c\in\mathbb{R}}\,\right\}}$$

(3) $${ W=\left\{\left.\,{\begin{bmatrix}c_1 \\ c_2 \\ 1 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c_1,c_2\in\mathbb{R}}\,\right\}}$$         (4) $${ W=\left\{\left.\,{\begin{bmatrix}c \\ -c \\ 1 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c\in\mathbb{R}}\,\right\}}$$

(5) $${ W=\left\{\left.\,{\begin{bmatrix}c_1 \\ c_2 \\ c_1-c_2 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c_1,c_2\in\mathbb{R}}\,\right\}}$$         (6) $${ W=\left\{\left.\,{\begin{bmatrix}c_1 \\ c_2 \\ c_3 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{c_1,c_2,c_3\in\mathbb{R},c_3>0}\,\right\}}$$         (7) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}\in \mathbb{R}^{3}}\,\,\right\vert\,\,{x_{1}+2x_{2}-x_{3}=0}\,\right\}}$$         (8) $${ W=\left\{\left.\,{\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}\in\mathbb{R}^3}\,\,\right\vert\,\,{x_1+x_2+x_3=0}\,\right\}}$$

(9) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}\in \mathbb{R}^{3}}\,\,\right\vert\,\,{x_{1}+x_{2}+x_{3}=1}\,\right\}}$$         (10) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...athbb{R}^{3}}\,\,\right\vert\,\,{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\geq0}\,\right\}}$$     (11) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...{array}{r} 3x_{1}-x_{2}+5x_{3}\ge0 \\ x_{2}-2x_{3}\ge0 \end{array}}\,\right\}}$$     (12) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...{1}-x_{2}+5x_{3}\ge1 \\ x_{2}-2x_{3}\ge1 \\ x_{3}\ge1 \end{array}}\,\right\}}$$

(13) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...x_{2} \\ x_{3} \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \end{bmatrix}}\,\right\}}$$

(14) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...d{bmatrix}= \begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix}\neq\vec{0}}\,\right\}}$$

(15) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...} \\ x_{3} \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}}\,\right\}}$$

(16) $${ W=\left\{\left.\,{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end... ...}= \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}\neq\vec{0}}\,\right\}}$$

問 3.45 (部分空間)   ベクトル空間 $\mathbb{R}[x]_3$ において， 部分集合 $W$ は部分空間であるか述べよ．

(1) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(x)=0}\,\right\}}$$          (2) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(x)=1}\,\right\}}$$

(3) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(0)=0,\,f(1)=0}\,\right\}}$$      (4) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(0)=1,\,f(1)=1}\,\right\}}$$      (5) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(3)=0,\,f(2)=0}\,\right\}}$$      (6) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(0)\geq0,\,f(1)\geq0}\,\right\}}$$      (7) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(0)\geq1,\,f(1)\geq1}\,\right\}}$$      (8) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(x)\ge0}\,\right\}}$$

(9) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(x)\ge1}\,\right\}}$$      (10) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{\left(f(1)\right)^2=0}\,\right\}}$$

(11) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f'(x)=0}\,\right\}}$$      (12) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f'(x)=1}\,\right\}}$$

(13) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(1)=0,\,f'(1)=0}\,\right\}}$$

(14) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(1)=0,\,f'(1)=0,\,f''(1)=0}\,\right\}}$$

(15) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(1)=1,\,f'(1)=1,\,f''(1)=1}\,\right\}}$$

(16) $${W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f'(3)=0,\,f(1)=0}\,\right\}}$$

(17) $${ W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f(0)+f'(1)=0,\,f'(0)+f''(1)=0}\,\right\}}$$

(18) $${W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_{3}}\,\,\right\vert\,\,{f''(x)-2xf'(x)=0}\,\right\}}$$

平成20年2月2日