91影视

Consider the set of all real linear space $C^2$.

The set $\left\{1,t,t_2,...,t_n\right\}$is linear independent set of V if there exist constant ${\mathit{a}}_{\mathbf{i}}\mathbf{}\mathbf{鈭�}\mathbf{}\mathit{R}$such that ${\mathit{a}}_{\mathbf{0}}\mathbf{+}{\mathit{a}}_{\mathbf{1}}{\mathit{t}}_{\mathbf{1}}\mathbf{+}{\mathit{a}}_{\mathbf{2}}{\mathit{t}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{a}}_{\mathbf{n}}{\mathit{t}}_{\mathbf{n}}\mathbf{=}\mathbf{-}\mathbf{}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}{\mathit{a}}_{\mathbf{0}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{=}{\mathit{a}}_{\mathbf{2}}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}{\mathit{a}}_{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}$where .

Any point $\left(z_1,z_2\right)鈭�C^2$is defined as follows.

role="math" localid="1659412444478" $\left(z_1,z_2\right)=\left(a_1+i{b}_1,a_2+i{b}_2\right)$

Simplify the equation $\left(z_1,z_2\right)=\left(a_1+i{b}_1,a_2+i{b}_2\right)$as follows.

$$\begin{gathered} \left(z_1,z_2\right)=\left(a_1+i{b}_1,a_2+i{b}_2\right) \\ \left(z_1,z_2\right)=\left(a_1,0)+(i{b}_1,0)+(0,a_2)+(0,i{b}_2\right) \\ \left(z_1,z_2\right)=a_1(1,0)+b_1(i,0)+a_2(0,1)+b_2(0,i) \end{gathered}$$

From the equation, the set $\left\{\left(1,0\right),\left(i,0\right),\left(0,1\right),\left(0,i\right)\right\}\mathrm{span}{\mathrm{C}}^2$.

Compare the equation $a_1\left(1,0\right)+b_1\left(i,0\right)+a_2\left(0,1\right)+b_2\left(0,i\right)=0$both sides as follows.

$$\begin{gathered} a_1\left(1,0\right)+b_1\left(i,0\right)+a_2\left(0,1\right)+b_2\left(0,i\right)=0 \\ {a}_1\left(1,0\right)+b_1\left(i,0\right)+a_2\left(0,1\right)+b_2\left(0,i\right)=\left(0\right)\left(1,0\right)+\left(0\right)\left(i,0\right)+\left(0\right)\left(0,1\right)+\left(0\right)\left(0,i\right) \\ {a}_i=0 \\ {b}_i=0 \end{gathered}$$

By the definition of linear independence, the subset role="math" localid="1659413116894" $\left\{\left(1,0\right),\left(i,0\right),\left(0,1\right),\left(0,i\right)\right\}$is L.I.

Therefore, the set of all real linear space $C^2$has dimension 4 and spanned by $span\left\{\left(1,0\right),\left(i,0\right),\left(0,1\right),\left(0,i\right)\right\}$.