91影视

Consider a basis of $\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}}\}$and role="math" localid="1660116504622" $\{{\stackrel{鈫�}{\mathrm{w}}}_1,{\stackrel{鈫�}{\mathrm{w}}}_2,...,{\stackrel{鈫�}{\mathrm{w}}}_{\mathrm{q}}\}$ for$V^{鈯�}$ in${\mathrm{鈩�}}^n{\mathrm{鈩�}}^n$ .

Assumerole="math" localid="1660116499579" $\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}},{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,....,{\stackrel{鈫�}{w}}_q\}$ and$\stackrel{鈫�}{v}鈭�{\mathrm{鈩�}}^n$ .

As ${\mathrm{鈩�}}^n=v+v^{鈯�}{\mathrm{鈩�}}^n=v+v^{鈯�}$and$\left\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}}\right\},\{,{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,....,{\stackrel{鈫�}{w}}_q\}$ are basis of V and$V^{鈯�}$ respectively.

There exists constant$a_i,b_i鈭�\mathrm{鈩�}$such that$\underset{i=1}{\overset{p}{鈭�}}{a}_i{\stackrel{鈫�}{v}}_i+\underset{i=1}{\overset{q}{鈭�}}{b}_i{\stackrel{鈫�}{w}}_i=v$ .

Therefore, the point$\stackrel{鈫�}{v}鈭�B$ .

Assume $\underset{i=1}{\overset{p}{鈭�}}{a}_i{\stackrel{鈫�}{v}}_i+\underset{i=1}{\overset{q}{鈭�}}{b}_i{\stackrel{鈫�}{w}}_i=0$where$\underset{i=1}{\overset{p}{鈭�}}{a}_i{\stackrel{鈫�}{v}}_i鈭�V$ and$\underset{i=1}{\overset{q}{鈭�}}{b}_i{\stackrel{鈫�}{w}}_i=0$ .

As$V鈭�V^{鈯�}=\left\{0\right\}$ implies$\underset{i=1}{\overset{p}{鈭�}}{a}_i{\stackrel{鈫�}{v}}_i=0$ and$\underset{i=1}{\overset{q}{鈭�}}{b}_i{\stackrel{鈫�}{w}}_i=0$ .

By the definition of linear independency, the value of the coefficients$a_i=0$ and$b_i=0$ for all$i$ .

If$\underset{i=1}{\overset{p}{鈭�}}{a}_i{\stackrel{鈫�}{v}}_i+\underset{i=1}{\overset{q}{鈭�}}{b}_i{\stackrel{鈫�}{w}}_i=0$ implies$a_i=0$ and$b_i=0$ , by the definition of linear independence the setrole="math" localid="1660117205931" $B=\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}},{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,....,{\stackrel{鈫�}{w}}_q\}$ is linear independence.

Hence, the set $\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}},{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,....,{\stackrel{鈫�}{w}}_q\}$spanned ${\mathrm{鈩�}}^n{\mathrm{鈩�}}^n$if $\{{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_2,....,{\stackrel{鈫�}{\mathrm{v}}}_{\mathrm{p}}\}$is basis of V and$\{{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,....,{\stackrel{鈫�}{w}}_q\}$ is basis for$V^{鈯�}$ .