Chapter 1 - Linear Equations in Linear Algebra - 1.8 Exercises - Page 71: 34

See solution

u and v are linearly independent, but T(u) and T(v) are linearly dependent. This means there are some nonzero scalars c and d such that cT(u)+dT(v)=0. However, because $au+bv\ne 0$ for any nonzero scalars a and b, the linear transformation must map onto the zero vector, showing that it maps nonzero vectors onto the zero vector.