## Can I say range $T =$ dim null $S$ + range $S$ if $S \in \mathcal{L}(V,W)$, and $T \in \mathcal{L}(U,V)$?

Can I say 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:

range $$T =$$ dim null $$S$$ + range $$S$$ (from S)

dim $$U$$ = dim null $$T$$ + range $$T$$ (from T)

dim $$U$$ = dim null $$ST$$ + range $$ST$$ (from $$ST \in \mathcal{L}(U,W)$$)

then

dim null $$T$$ + range $$T$$ = dim null $$ST$$ + range $$ST$$

dim null $$T$$ + dim null $$S$$ + range $$S$$ = dim null $$ST$$ + range $$ST$$

dim null $$ST$$ = dim null $$T$$ + dim null $$S$$ + range $$S$$ – range $$ST$$

The trial seems not working. Maybe range $$T =$$ dim null $$S$$ + range $$S$$ is wrong in the first place.