3.33 演習問題～解空間

問 3.135 (解空間) 次の解空間の基底と次元を求めよ．ただし，(12), (13), (19)では $\dim(W)\ge1$ となるよう

の値を定めよ．

(1) $\displaystyle{W=\left\{\left.\,{\vec{x}\in\mathbb{R}^{n}}\,\,\right\vert\,\,{A\vec{x}=\vec{0}\,\text{ただし $\mathrm{rank}\,(A)=r$}}\,\right\}}$ (2) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^2 \left\vert 2x_{1}-x_{2}=0 \right.\right\}}$

(3) $\begin{displaymath}\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^2 \left\vert \be... ...} 2x_{1}-x_{2}=0 \\ x_{1}+x_{2}=0 \end{array}\right.\right\}}\end{displaymath}$ (4) $\begin{displaymath}\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^2 \left\vert \be... ... 2x_{1}-x_{2}=0 \\ -2x_{1}+x_{2}=0 \end{array}\right.\right\}}\end{displaymath}$

(5) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^2 \quad\left\vert\quad \begin{bma... ...だし $a_{11}a_{22}-a_{12}a_{21}=0$, $a_{11}\neq0,a_{22}\neq0$} \right.\right\}}$

(6) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^2 \quad\left\vert\quad \begin{bma... ...� $a_{11}a_{22}-a_{12}a_{21}\neq0$, $a_{11}\neq0,a_{22}\neq0$} \right.\right\}}$

(7) $\displaystyle{W= \left\{\left.\,{\vec{x}\in\mathbb{R}^3}\,\,\right\vert\,\,{x_1+x_2+x_3=0}\,\right\}}$ (8) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatrix} 1 & 2 & -1 \\ 3 &-3 & 2 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(9) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatrix} 1 & 0 & -1 \\ 2 &-1 & 3 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (10) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatrix} 1 & 1 & -1 \\ 2 & 2 & -2 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(11) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatr... ...& 0 \\ 0 & 3 & -1 \\ 2 & -2 & 0 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (12) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatr... ...-a \\ 2 & -3 & 1 \\ 3 & -1 & -2 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(13) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatr... ... -1 \\ 3 & a & -2 \\ 1 & 5a & 0 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (14) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^3 \,\,\left\vert\,\, \begin{bmatr... ...& 0 \\ 0 & 3 & -1 \\ 1 & 2 & -1 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(15) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^4 \,\,\left\vert\,\, \begin{bmatr... ...1 & -2 & -3 & 4 \\ 0 & 2 & 2 & 0 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (16) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^4 \,\,\left\vert\,\, \begin{bmatr... ...1 & 1 & -1 & 1 \\ 3 & 1 & 2 & -1 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(17) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^4 \,\,\left\vert\,\, \begin{bmatr... ... & 2 & -2 & -1 \\ 3 & 2 & 6 & 11 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (18) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^4 \,\,\left\vert\,\, \begin{bmatr... ...2 & -2 & 0 & 2 \\ 1 & -2 & 0 & 2 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(19) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^4 \left\vert \begin{bmatrix} 1 & ... ... & -6 & 2 & 13 \\ 2 & -4 & 1 & 9 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$ (20) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^5 \left\vert \begin{bmatrix} 1 & ... ... 1 & 0 & 2 \\ 2 & 1 & 2 & -1 & 5 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(21) $\displaystyle{W= \left\{\vec{x}\in\mathbb{R}^5 \,\,\left\vert\,\, \begin{bmatr... ... & 1 & -5 \\ 3 & 1 & 4 & -7 & 10 \end{bmatrix}\vec{x}=\vec{0} \right.\right\}}$

(22) $W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_3}\,\,\right\vert\,\,{f(1)=0,\,\,f'(1)=0}\,\right\}$ (23) $W=\left\{\left.\,{f(x)\in\mathbb{R}[x]_3}\,\,\right\vert\,\,{f(1)=0,\,\,f(-1)=0}\,\right\}$

3.33 演習問題 ～ 解空間

3.33 演習問題～解空間