Sure. If T is not one-to-one, then two different (possibly linearly independent) vectors could be mapped by T to the same vector. That is, T(x) = T(y). It follows that T(x) - T(y) = 0, a linear combination of T(x) and T(y) that equals zero, and the coefficients are nonzero. Therefore, T(x) and T(y) are linearly dependent. QED.