Take two linear transformations T from V to W and S from W to U. I want to show that the dimension of the image of their composite SoT from V to U is 'smaller' than or equal to the dimension of the image of S.
Attempts:
I've been trying to substitute different values into the theorem that says for a linear transformation T from V to W, dim(v)=dim(ker(T))+dim(im(T)). I've also been using the facts that kernel and images are subspaces of V and W, respectively. Notably, dim(ker(T)) is less than or equal to dim(V) and dim(Im(T)) is less than or equal to dim(W).
Best Answer
You can easily show that $Im(S \circ T) \subset Im(S)$. The result you need then follows immediately.