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## Homework Statement

Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that

dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)]

## Homework Equations

(*) dim[Ker(S)] = dim(U) - dim[Im(T)]

(**) dim[Ker(T)] = dim(V) - dim[Im(S)]

## The Attempt at a Solution

I know that ST is in L(U, W), so dim[Ker(ST)] = dim(U) - dim[Im(ST)]. So now I need to show:

dim(U) - dim[Im(ST)] <= dim(U) + dim(V) - dim[Im(T)] - dim[Im(S)]

But that just boils down to showing that dim[Im(ST)] is greater than dim[Im(T)] + dim[Im(S)]. It seems like I didn't really get anywhere. What am I missing? Thanks for your help.