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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 span $\left({\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,.......,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{m}}\right)$should satisfy all the above conditions.

The first condition is,

$\left({\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,.......,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{m}}\right)$can be written as,

$鈭�\stackrel{鈫�}{0}鈭�W$

The first condition holds good.

The second condition is,

$\stackrel{鈫�}{0}=0\left({\stackrel{鈫�}{v}}_1+{\stackrel{鈫�}{v}}_2\right)$

${\stackrel{鈫�}{v}}_1+{\stackrel{鈫�}{v}}_2鈭�W$

The second condition holds good.

The third condition is,

The linear combination of span $\left({\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,.........,{\stackrel{鈫�}{v}}_m\right)$is still a part of the span of span

The third condition also holds good.

The span $\left({\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,.........,{\stackrel{鈫�}{v}}_m\right)$is a subspace of ${\mathrm{鈩�}}^n$.