the first statement is false. A couterexample is T=0 for all vectors in V.

the second statement is true. Because a linear transformation preserve addition and multiplication by scalar, so if v_1,...v_n are linear dependent,that is, there exist nubmers a_1,...a_n (not all 0) in F such that the linear combination of v_1,...v_n is 0, thus the image of this linear combination under T is 0 in W, that is T(v_1), ...T(v_n) is dependent linear.

To summarize, A linear transform T preserve the linear independency iff T is injective( T is invertable).