# find the standard matrix for the linear operator t defined by the formula T(x,y,z)=(x-2y ,2x+y)

find the standard matrix for the linear operator t defined by the formula $T\left(x,y,z\right)=\left(x-2y,2x+y\right)$
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Here the linear transformation is given by $T\left(x,y,z\right)=\left(x-2y,2x+y\right).$ The standard basis of ${R}^{3}isB2=\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)\right\}$
$T\left(1,0,0\right)=\left(1,2\right)=1\left(1,0\right)+2\left(0,1\right)$
$T\left(0,1,0\right)=\left(-2,1\right)=-2\left(1,0\right)+1\left(0,1\right)$
$T\left(0,0,1\right)=\left(0,0\right)=0\left(1,0\right)+0\left(0,1\right).$
Therefore the standard matrix of T is $\left[T\right]\left[B2,B3\right]=\left[1,2,-2,1,0,0\right].$