91影视

A subset of the vector space ${\mathrm{鈩�}}^n$is called a (linear) subspace of if it has the following three properties:

a. W contains the zero vector in ${\mathrm{鈩�}}^n$.

b. W is closed under addition: If ${\stackrel{鈫�}{w}}_1$and ${\stackrel{鈫�}{w}}_2$are both in W , then so is ${\stackrel{鈫�}{w}}_1+{\stackrel{鈫�}{w}}_2$.

c. W is closed under scalar multiplication: If $\stackrel{鈫�}{w}$is in W and k is an arbitrary scalar, then $k\stackrel{鈫�}{w}$is in W .

The set W should satisfy all the above conditions.

$W=\left\{\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x鈮�y鈮�z\right\}$

The first condition is,

$W=\left\{\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x鈮�y鈮�z\right\}$

$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]:0鈮�0鈮�0$

The first condition holds good.

The second condition is,

$$\begin{gathered} {\stackrel{鈫�}{w}}_1=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x鈮�y鈮�z,{\stackrel{鈫�}{w}}_2=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]:a鈮�b鈮�c \\ {\stackrel{鈫�}{w}}_1+{\stackrel{鈫�}{w}}_2=\left\{\left[\begin{array}{c}x+a\\ y+b\\ z+c\end{array}\right]:x+a鈮�y+b鈮�z+c\right\} \end{gathered}$$

The second condition holds good.

The third condition is,

If $k>0$

$k{\stackrel{鈫�}{w}}_1=\left\{\left[\begin{array}{c}kx\\ ky\\ kz\end{array}\right]:kx鈮�ky鈮�kz\right\}$

If k<0

$k{\stackrel{鈫�}{w}}_1=\left\{\left[\begin{array}{c}kx\\ ky\\ kz\end{array}\right]:kx鈮�ky鈮�kz\right\}$

The third condition does not hold good.

$W=\left\{\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x鈮�y鈮�z\right\}$ is not a subset of ${\mathrm{鈩�}}^3$ .