Question

Check whether T(x1,x2,x3)=(x1+x2/x3,x3) define linear transformation from R3 to R2

Answer 1

Let us check whether $T(x_1,x_2,x_3)=(x_1+\frac{x_2}{x_3},x_3)$ define a linear transformation from $\R^3$ to $\R^2.$

Taking into account that

$T(x_1,x_2,x_3)+T(y_1,y_2,y_3)=(x_1+\frac{x_2}{x_3},x_3)+(y_1+\frac{y_2}{y_3},y_3)=(x_1+y_1+\frac{x_2}{x_3}+\frac{y_2}{y_3},x_3+y_3)$

and $T(x_1+y_1,x_2+y_2,x_3+y_3)=(x_1+y_1+\frac{x_2+y_2}{x_3+y_3},x_3+y_3)\ne T(x_1,x_2,x_3)+T(y_1,y_2,y_3),$

we conclude that $T(x_1,x_2,x_3)=(x_1+\frac{x_2}{x_3},x_3)$ does not define a linear transformation from $\R^3$ to $\R^2.$