91影视

A subset W of the vector space ${\mathrm{鈩�}}^n$is called a (linear) subspace of ${\mathrm{鈩�}}^n$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 subspaces of V and W should satisfy the above conditions.

The first condition is,

The subspaces of V and W belong to ${\mathrm{鈩�}}^n$can be written as,

$$\begin{gathered} 鈬�0鈭�V \\ 鈬�0鈭�W \\ 鈭�0鈭�V鈭�W \end{gathered}$$

The first condition holds good.

The second condition is,

$$\begin{gathered} x,y鈭�V鈭�W \\ 鈬�x,y鈭�V \\ 鈬�x,y鈭�W \\ 鈭�x+y鈭�V鈭�W \end{gathered}$$

The second condition holds good.

The third condition is,

$$\begin{gathered} x鈭�V鈭�W,k鈭�\mathrm{鈩�} \\ 鈬�x鈭�V鈬�kx鈭�V \\ 鈬�x鈭�W鈬�kx鈭�W \\ 鈭�kx鈭�V鈭�W \end{gathered}$$

The third condition also holds good.

Therefore, $V鈭�W$is a subspace of ${\mathrm{鈩�}}^n$.

The second condition is,

$$\begin{gathered} x,y鈭�V鈭�W \\ 鈬�x,y鈭�V \\ 鈬�x,y鈭�V \\ 鈭�x+y鈭�V鈭�W \end{gathered}$$

The second condition does not hold good.

Therefore, $V鈭�W$is not a subspace of ${\mathrm{鈩�}}^n$.

a. $V鈭�W$is a subspace of ${\mathrm{鈩�}}^n$

b. $V鈭�W$is not a subspace of ${\mathrm{鈩�}}^n$.