# A question on kernel and image of a composition of linear transformation.

Let $$V,W,U$$ be vector spaces over $$\mathbb{F}$$ such that $$V$$ and $$W$$ are finite dimensional. Let $$f:V\to W$$ and $$g:W\to U$$ be linear transformations over $$\mathbb{F}$$.

a) Let $$h:f(V)\to U$$ such that $$h(w):=g(w)$$ for all $$w\in f(V)$$. How are Im$$(g\circ f)$$ and Im$$(h)$$ related and why? How are ker$$(h)$$ and ker$$(g)$$ related and why?

b) Show that rank$$(f)\leq$$rank$$(g\circ f)$$+nul$$(g)$$.

For item a, what I did was to use the fact that images are subspaces of the codomain of a linear transformation hence it kinda sounds like the images are the same, but I actually have no idea how to move on nor if the why I think of it is correct. For item b, I haven't had the luck yet. Can I ask for help for both items? A hint would be so appreciated, I still want to work out the proof on my own. I just want a headstart. Thank you.

For a, your guess that the images are the same is correct. Keeping in mind the definition of the image of a linear transformation, it suffices to note that for every $$w \in f(V)$$, $$h(w)$$ can be written in the form $$g(f(v))$$ for some vector $$v \in V$$.
In order to relate the kernel of these two maps, note that if $$h(w) = 0$$ holds for some $$w \in f(V) \subset W$$, then it follows that $$g(w) = 0$$.
For b, recall that the rank of $$f$$ is the dimension of $$f(V)$$. Now, apply the rank-nullity theorem to the map $$h$$.