Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 300: 23

$T(1,4)=(-3,5)\\ T(-2,1)=(-3,-1)$

Observe that $(1,4)=1(1,0)+4(0,1)$ Since, $T$ is a linear transformation, $T(1,4)\\=T[1(1,0)+4(0,1)]\\=1T(1,0)+4T(0,1)\\ =(1,1)+4(-1,1)\\ =(-3,5)$ Now, observe that $(-2,1)=(-2)(1,0)+1(0,1)$ Since, $T$ is a linear transformation, $T(-2,1)\\=T[(-2)(1,0)+1(0,1)]\\=(-2)T(1,0)+1T(0,1)\\ =(-2)(1,1)+1(-1,1)\\ =(-3,-1)$