# text{Let} overrightarrow{e_1}, overrightarrow{e_2}, overrightarrow{e_3} text{be standard unit vectors along the coordinate axes in} R^3. text{Let S an

$S\left(over\to {\left\{e\right\}}_{1}\right)=T\left(over\to {\left\{e\right\}}_{1}\right),S\left(over\to {\left\{e\right\}}_{2}\right)=T\left(over\to {\left\{e\right\}}_{2}\right),S\left(over\to {\left\{e\right\}}_{3}\right)=T\left(over\to {\left\{e\right\}}_{3}\right)$
then
$S\left(over\to \left\{x\right\}\right)=T\left(over\to \left\{x\right\}\right)$
for any $over\to \left\{x\right\}\in {R}^{3}$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Brighton

Then, using properties of linear transformation,
$⇒S\left(over\to \left\{x\right\}\right)$

We know that,
$\left[S\left(over\to {\left\{e\right\}}_{1}\right)=T\left(over\to {\left\{e\right\}}_{1}\right),S\left(over\to {\left\{e\right\}}_{2}\right)=T\left(over\to {\left\{e\right\}}_{2}\right),S\left(over\to {\left\{e\right\}}_{3}\right)=T\left(over\to {\left\{e\right\}}_{3}\right)\right]$

$⇒T\left(over\to \left\{x\right\}\right)$
Thus, proved $S\left(over\to \left\{x\right\}\right)=T\left(over\to \left\{x\right\}\right)$