Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 300: 10

Not a linear transformation.

The function $T(x,y)=(x^2,y)$ is not a linear transformation. This is because, one can see that $$T((x_1,y_1)+(x_2,y_2))=T(x_1+x_2,y_1+y_2)=((x_1+x_2)^2,y_1+y_2)$$ and $$T(x_1,y_1)+T(x_2,y_2)=(x_1^2,y_1)+(x_2^2,y_2)=(x_1^2+x_2^2,y_1+y_2).$$ Since, generally, $(x_1+x_2)^2\neq x_1^2+x_2^2 $, then $$T((x_1,y_1)+(x_2,y_2))\neq T(x_1,y_1)+T(x_2,y_2).$$