Projection - linear transformation

V = V1 V2 is a direct sum of two subspaces V1 and V2.

T is the projection transformation of V onto V1, parallel to V2. Find ImT and KerT.

This is a classic question and the answer is ImT = V1 and kerT = V2

But, as usual, I'm having trouble understanding why. Can someone tell me if this is correct:

since V = V1 V2, for every v V, v = v1 + v2.

T(v) = T(v1+v2) = v1, so ImT = v1

kerT is when the image is equal to 0 -- T(v) = T(v1+v2)=v1 -- but v1 is equal to zero, so T(v2) = 0.

which makes kerT = V2.

Is this the correct way to look at it??