Suppose $F: V \to U$ is linear and $k$ is a nonzero scalar. Prove that the maps $F$ and $kF$ have the same kernel and the same image.

Question

Any help starting this proof or proving the above is appreciated. I know how to prove for the kernel now but not sure about proving for the image.

I've been trying to prove for the image using the definition of the image of F and one of two conditions that make a mapping linear (that is scalar multiplication which says for any scalar k, and v (vector) is an element of V (the vector space), we have F(kv) = kF(v)).

Does anybody know any helpful definitions for proving the image?