Can I say dim range $T = $ dim null $S$ + range $S$ if $S \in \mathcal{L}(V,W)$, and $T \in \mathcal{L}(U,V)$?
The reason I asked this is because I am doing Q22 of section 3.B on Linear Algebra Done Right.
Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V,W)$, and $T \in \mathcal{L}(U,V)$. Prove that $$\text{dim null }ST \leq \text{dim null }S + \text{dim null }T$$
My failed trail is as follow:
dim range $T = $ dim null $S$ + dim range $S$ (from S)
dim $U$ = dim null $T$ + dim range $T$ (from T)
dim $U$ = dim null $ST$ + dim range $ST$ (from $ST \in \mathcal{L}(U,W)$)
then
dim null $T$ + dim range $T$ = dim null $ST$ + dim range $ST$
dim null $T$ + dim null $S$ + dim range $S$ = dim null $ST$ + dim range $ST$
dim null $ST$ = dim null $T$ + dim null $S$ + dim range $S$ - dim range $ST$
The trial seems not working. Maybe range $T = $ dim null $S$ + dim range $S$ is wrong in the first place.