91影视

If$\mathfrak{I}$be the basis of a subspace V of ${\mathrm{鈩�}}^n$then

${\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}+{\left[\stackrel{鈬赌}{y}\right]}_{\mathfrak{I}}$ for all vectors in $\stackrel{鈬赌}{\mathrm{x}}\mathrm{and}\stackrel{鈬赌}{\mathrm{y}}$V.

Let $\mathfrak{I}=\left\{{\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_2,...,{\stackrel{鈬赌}{v}}_n\right\}$be the basis of the subspace V of${\mathrm{鈩�}}^n$

Then any vectors $\stackrel{鈬赌}{x},\stackrel{鈬赌}{y}鈭�{\mathrm{鈩�}}^n$can be uniquely represented as a linear combinations of the vectors ${\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_2,...,{\stackrel{鈬赌}{v}}_n$. Then we have

localid="1660645060506" $$\begin{gathered} \stackrel{鈬赌}{x}=a_1{\stackrel{鈬赌}{v}}_1+a_2{\stackrel{鈬赌}{v}}_2+...+a_n{\stackrel{鈬赌}{v}}_n \\ and \\ \stackrel{鈬赌}{y}=b_1{\stackrel{鈬赌}{v}}_1+b_2{\stackrel{鈬赌}{v}}_2+...+b_n{\stackrel{鈬赌}{v}}_n \end{gathered}$$

localid="1660645555472" $鈬�{\left[\stackrel{鈬赌}{x}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_n\end{array}\right]\mathrm{and}{\left[\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ .\\ .\\ .\\ {\mathrm{b}}_{\mathrm{n}}\end{array}\right]$

Since we have

$$\begin{gathered} \stackrel{鈬赌}{x}=a_1{\stackrel{鈬赌}{v}}_1+a_2{\stackrel{鈬赌}{v}}_2+...+a_n{\stackrel{鈬赌}{v}}_n \\ and \\ \stackrel{鈬赌}{y}=b_1{\stackrel{鈬赌}{v}}_1+b_2{\stackrel{鈬赌}{v}}_2+...+b_n{\stackrel{鈬赌}{v}}_n \end{gathered}$$

Where

$$\begin{gathered} 鈬�\stackrel{鈬赌}{x}+\stackrel{鈬赌}{y}=a_1{\stackrel{鈬赌}{v}}_1+a_2{\stackrel{鈬赌}{v}}_2+...+a_n{\stackrel{鈬赌}{v}}_n\stackrel{鈬赌}{x}+b_1{\stackrel{鈬赌}{v}}_1+b_2{\stackrel{鈬赌}{v}}_2+...+b_n{\stackrel{鈬赌}{v}}_n+ \\ 鈬�\stackrel{鈬赌}{x}+\stackrel{鈬赌}{y}=(a_1+b_1{\stackrel{鈬赌}{)v}}_1+(a_2+b_2){\stackrel{鈬赌}{v}}_2+...(a_n+b_n){\stackrel{鈬赌}{v}}_n \\ 鈬�\stackrel{鈬赌}{x}+\stackrel{鈬赌}{y}=c_1{\stackrel{鈬赌}{v}}_1+c_2{\stackrel{鈬赌}{v}}_2+...+c_n{\stackrel{鈬赌}{v}}_n \\ \mathrm{Where}{\mathrm{c}}_{\mathrm{i}}={\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}},\mathrm{i}=1,2,3,...\mathrm{n}. \\ 鈬�{\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{\mathrm{c}}_1\\ {\mathrm{c}}_2\\ .\\ .\\ .\\ {\mathrm{c}}_{\mathrm{n}}\end{array}\right] \end{gathered}$$

We have

$$\begin{gathered} 鈬�{\left[\stackrel{鈬赌}{x}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_n\end{array}\right]\mathrm{and}{\left[\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ .\\ .\\ .\\ {\mathrm{b}}_{\mathrm{n}}\end{array}\right] \\ \mathrm{And} \\ 鈬�{\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{\mathrm{c}}_1\\ {\mathrm{c}}_2\\ .\\ .\\ .\\ {\mathrm{c}}_{\mathrm{n}}\end{array}\right] \\ {\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}{\mathrm{c}}_1\\ {\mathrm{c}}_2\\ .\\ .\\ .\\ {\mathrm{c}}_{\mathrm{n}}\end{array}\right]=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ .\\ a_n\end{array}\right]+\left[\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ .\\ .\\ .\\ {\mathrm{b}}_{\mathrm{n}}\end{array}\right] \\ 鈬�{\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}={\left[\stackrel{鈬赌}{x}\right]}_{\mathfrak{I}}+{\left[\stackrel{鈬赌}{y}\right]}_{\mathfrak{I}} \end{gathered}$$

If$$\begin{gathered} \mathfrak{I}=\left\{{\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_2,...,{\stackrel{鈬赌}{v}}_n\right\}\mathrm{be}\mathrm{the}\mathrm{basis}\mathrm{of}\mathrm{a}\mathrm{subspace}\mathrm{V}\mathrm{of}{\mathrm{鈩�}}^{\mathrm{n}},\mathrm{then}{\left[\stackrel{鈬赌}{\mathrm{x}}+\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}={\left[\stackrel{鈬赌}{\mathrm{x}}\right]}_{\mathfrak{I}}+{\left[\stackrel{鈬赌}{\mathrm{y}}\right]}_{\mathfrak{I}}\mathrm{for} \\ \mathrm{all}\mathrm{vectors}\stackrel{鈬赌}{\mathrm{x}}\mathrm{and}\stackrel{鈬赌}{\mathrm{y}}\mathrm{in}\mathrm{V}. \end{gathered}$$be