91影视

A transformation T from${\mathrm{鈩�}}^n$to${\mathrm{鈩�}}^n$is called linear if and only if

1. $T(\stackrel{鈫�}{x}+\stackrel{鈫�}{y})=T\left(\stackrel{鈫�}{x}\right)+T\left(\stackrel{鈫�}{y}\right)$for all vectors$\stackrel{鈫�}{x}and\stackrel{鈫�}{y}$in ${\mathrm{鈩�}}^n$.
2. localid="1661169981986" $T\left(k\stackrel{鈫�}{x}\right)=kT\left(\stackrel{鈫�}{x}\right)$for all vectors $\stackrel{鈫�}{x}$in ${\mathrm{鈩�}}^n$and all scalars k.

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

1. $[\stackrel{鈫�}{x}+\stackrel{鈫�}{y}{]}_{貜}=[\stackrel{鈫�}{x}{]}_{貜}+[\stackrel{鈫�}{y}{]}_{貜}$ for all vectors in V (1)
2. ${\left[k\stackrel{鈫�}{x}\right]}_{\mathrm{峥�}}=k{\left[\stackrel{鈫�}{x}\right]}_{貜}$for all vectors$\stackrel{鈫�}{x}$ in V and all scalars k. (2)

Since we have given

$T\left(\stackrel{鈫�}{x}\right)={\left[x\right]}_{貜}$ (3)

$$\begin{gathered} 鈬�T\left(\stackrel{鈫�}{x}+\stackrel{鈫�}{y}\right)={\left[\stackrel{鈫�}{x}+\stackrel{鈫�}{y}\right]}_{貜} \\ 鈬�T\left(\stackrel{鈫�}{x}+\stackrel{鈫�}{y}\right)=\left[\stackrel{鈫�}{x}\right]+{\left[\stackrel{鈫�}{y}\right]}_{貜}\left(from\left(1\right)\right) \\ 鈬�T\left(\stackrel{鈫�}{x}+\stackrel{鈫�}{y}\right)=T{\left[\stackrel{鈫�}{x}\right]}_{貜}+T{\left[\stackrel{鈫�}{y}\right]}_{貜}\left(from\left(3\right)\right) \end{gathered}$$

Now,

$$\begin{gathered} 鈭�T\left(\stackrel{鈫�}{x}\right)=\left[\stackrel{鈫�}{x}\right] \\ 鈬�T\left(k\stackrel{鈫�}{x}\right)={\left[k\stackrel{鈫�}{x}\right]}_{貜} \\ 鈬�T\left(k\stackrel{鈫�}{x}\right)=k{\left[\stackrel{鈫�}{x}\right]}_{貜}\left(from\left(2\right)\right) \\ 鈬�T\left(k\stackrel{鈫�}{x}\right)=kT{\left[\stackrel{鈫�}{x}\right]}_{貜}\left(from\left(2\right)\right) \end{gathered}$$

Since the transformation T from ${\mathrm{鈩�}}^n$to ${\mathrm{鈩�}}^n$such that $T\left(\stackrel{鈫�}{x}\right)={\left[x\right]}_{貜}$we have

1. $T(\stackrel{鈫�}{x}+\stackrel{鈫�}{y})=T\left(\stackrel{鈫�}{x}\right)+T\left(\stackrel{鈫�}{y}\right)$for all vectors $\stackrel{鈫�}{\mathrm{x}}\mathrm{and}\stackrel{鈫�}{\mathrm{y}}$in ${\mathrm{鈩�}}^n$.

2.localid="1661169997570" $T\left(k\stackrel{鈫�}{x}\right)=kT\left(\stackrel{鈫�}{x}\right)$for all vectors $\stackrel{鈫�}{x}$in ${\mathrm{鈩�}}^n$and all scalars k.

This shows that the transformation T from ${\mathrm{鈩�}}^n$to ${\mathrm{鈩�}}^n$such that $T\left(\stackrel{鈫�}{x}\right)=[x{]}_{貜}$is linear.