Determine which of the following transformations are linear transformations
A. The transformation $T_1$ defined by $T_1(x_1,x_2,x_3)=(x_1,0,x_3)$
B. The transformation $T_2$ defined by $T_2(x_1,x_2)=(2x_1−3x_2,x_1+4,5x_2)$.
C. The transformation $T_3$ defined by $T_3(x_1,x_2,x3)=(x_1,x_2,−x_3)$
D. The transformation $T_4$ defined by $T_4(x_1,x_2,x3)=(1,x_2,x_3)$
E. The transformation $T_5$ defined by $T_5(x_1,x_2)=(4x_1−2x_2,3|x_2|)$.
I believe that it could be A and E. How can I determine this? If someone could show me one I could figure out the rest.
Best Answer
To test whether T is a linear transformation, you need to check that for some vectors $a$ and $b$ and some constant $c$
$$T(a + b) = T(a) +T(b)$$ $$T(ca) = cT(a)$$ $$T(0) = 0$$ So for example,
A. $T(x_1,x_2,x_3)=(x_1,0,x_3)$ $$T(x_1+y_1,x_2+y_2,x_3+y_3)=(x_1+y_1,0(x_2+y_2),x_3+y_3)=T(x_1,0,x_3)+T(y_1,0,y_2)$$$$T(cx_1,cx_2,cx_3)=T(cx_1,(c)0,cx_3)=cT(x_1,0,x_3)$$ $$T(0,0,0)=0$$ B. $T(x_1,x_2)=(2x_1−3x_2,x_1+4,5x_2)$ $$T(x_1+y_1,x_2+y_2)=(2(x_1+y_1)−3(x_2+y_2),(x_1+y_1)+4,5(x_2+y_2))=(2x_1+2y_1−3x_2-3y_2,x_1+y_1+4,5x_2+5y_2)$$$$T(x_1,x_2)+T(y_1,y_2)=(2x_1−3x_2,x_1+4,5x_2)+(2y_1−3y_2,y_1+4,5y_2)=(2x_1−3x_2+2y_1-3y_2,x_1+y_1+8,5x_2+5y_2)\not=T(x_1+y_1,x_2+y_2)$$ So B is not a linear transformation.