Craig French

2022-02-14

Let T be a linear transformation from such that . Find
$T\left(6,1\right)=?$
$T\left(-5,1\right)=?$

meldElafellrbo

Concept: property of linear transformation from states that $T\left(v1+v2\right)=T\left(v1\right)+T\left(v2\right)$ where T is linear transformation and v1, v2 are vectors from ${R}^{n}$
$T\left(a\cdot V1\right)=aT\left(V1\right)$ where a is scalar, V1 is vector
given $T\left(1,0\right)=\left(1,1\right)$
$T\left(0,1\right)=\left(-1,1\right)$

$T\left(6,1\right)=T\left(6+0,0+1\right)=T\left(\left(6,0\right)+\left(0,1\right)\right)$
using properties of linear transformation we get
$=T\left(6,0\right)+T\left(0,1\right)$
$=T\left(6\left(1,0\right)\right)+T\left(0,1\right)$
$=6T\left(1,0\right)+T\left(0,1\right)$
$=6\left(1,1\right)+\left(-1,1\right)$
$=\left(6,6\right)+\left(-1,1\right)$
$=\left(6-1,6+1\right)=\left(5,7\right)$
linear transformation of (6,1) is (5,7)
$T\left(-5,1\right)=T\left(-5+0,0+1\right)=T\left(\left(-5,0\right)+\left(0,1\right)\right)$
using properties of linear transformation we get
$=T\left(-5,0\right)+T\left(0,1\right)$
$=T\left(-5\left(1,0\right)\right)+T\left(0,1\right)$
$=-5T\left(1,0\right)+T\left(0,1\right)$
$=-5\left(1,1\right)+\left(-1,1\right)$
$=\left(-5,-5\right)+\left(-1,1\right)$
$=\left(-5-1,-5+1\right)=\left(-6,-4\right)$
linear transformation of (-5,1) is (-6,-4)

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