# vector spaces – Prove \$dim(im(g circ f)) ≤ min{dim(im g), dim(im f)}\$ and \$dim(ker(g circ f)) ≥ dim(ker f)\$

For the finite-dimensional Vector spaces U, V and W with $$f: U → V$$ and $$g: V → W$$ show the following:

1. dim$$($$im $$(g circ f)) ≤$$min$${$$dim$$($$im $$g),$$dim$$($$im $$f)}$$
2. dim$$($$ker $$(g circ f)) ≥$$dim$$($$ker $$f)$$

Im aware of a possible solution to a similar proof to (1) by using the rank nullity problem to show $$dim Ker(f+g) leq dim (Ker f cap Ker g) + dim (Im f cap Im g)$$, im unsure on how to apply that here though.

For two i got $$dim(ker(f circ g)=dim(V)−dim fg(V)≥dim V −dim f(U)=dim(ker f)$$
but im not 100% sure about that.