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Let $\stackrel{鈫�}{x}鈭�{\mathrm{鈩�}}^n.$ Then

$$\begin{gathered} \stackrel{鈫�}{x}=\underset{i=1}{\overset{n}{鈭�}}{a}_i{v}_i \\ \text{where}{a}_1,a_2,...,a_n\text{are unique scalars} \end{gathered}$$

Define a map

$$\begin{gathered} T:{\mathrm{鈩�}}^n鈫�{\mathrm{鈩�}}^n\text{by} \\ \text{T(}x)=\underset{i=1}{\overset{n}{鈭�}}{a}_i{w}_i\text{.} \end{gathered}$$

Suppose that$u,v鈭�{\mathrm{鈩�}}^n\text{and}d鈭�F.$

Then we may write

$\stackrel{鈫�}{u}=\underset{i=1}{\overset{n}{鈭�}}{b}_i{v}_i\text{and}\stackrel{鈫�}{v}=\underset{i=1}{\overset{n}{鈭�}}{c}_i{v}_i$

for scalars$b_1,b_2,....,b_n\text{and}{c}_1,c_2...,c_n$ .

Thus,

$du+v=\underset{i=n}{\overset{n}{鈭�}}(d{b}_i+c_i)v_i$

$鈬�T(du+v)=\underset{i=1}{\overset{n}{鈭�}}(d{b}_i+c_i)w_i.$

$鈬�T(du+v)=d\underset{i=1}{\overset{n}{鈭�}}(b_i{w}_i)+\underset{i=n}{\overset{n}{鈭�}}{c}_i{w}_i=dT(u)+T(v).$

Clearly,$T\left(v_i\right)=w_i\text{for}i=1,2,...,n.$

Suppose $U:{\mathrm{鈩�}}^n鈫�{\mathrm{鈩�}}^n$is linear and$U\left(v_I\right)=w_I\text{for}i=1,2,...,n.$

Then for $\stackrel{鈫�}{x}鈭�{\mathrm{鈩�}}^n$with

$$\begin{gathered} \stackrel{鈫�}{x}=\underset{i=1}{\overset{n}{鈭�}}{a}_i{v}_i \\ \text{where}{a}_1,a_2,...,a_n\text{are unique scalars.} \end{gathered}$$

We have

$U\left(x\right)=\underset{i=1}{\overset{n}{鈭�}}{a}_{i}U\left(v_i\right)=\underset{i=1}{\overset{n}{鈭�}}{a}_i{w}_i=T\left(x\right)\text{.}$

Hence, U =T.

If ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3,...,{\stackrel{鈫�}{v}}_n\text{and}{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,{\stackrel{鈫�}{w}}_3,...,{\stackrel{鈫�}{w}}_n$are two bases of ${\mathrm{鈩�}}^n$ , then there exists a unique linear transformation T from${\mathrm{鈩�}}^n\text{to}{\mathrm{鈩�}}^n$ such that $T\left({\stackrel{鈫�}{v}}_1\right)={\stackrel{鈫�}{w}}_1,T\left({\stackrel{鈫�}{v}}_2\right)={\stackrel{鈫�}{w}}_2,...,T\left({\stackrel{鈫�}{v}}_n\right)={\stackrel{鈫�}{w}}_n$.