When two linear transformations are preformed one after another, the combined effect may not always be a linear transformation. Is it true?
What I thought is below.
Let $T$ and $U$ are linear transformation which is $T:\Bbb{R}^n\to \Bbb{R}^m$, $U:\Bbb{R}^m\to \Bbb{R}^l$ then they have to satisfy the condition of linear transformation, so I can write as below.
$T(cu+dv) =cT(u)+dT(v)$
$U(eu+fv)=eU(u)+fU(v)$
for some constants $c,d,e,v$.
If $T$ is first used and after $U$,
$U(T(cu+dv))=U(cT(u)+dT(v))=cU(T(u))+dU(T(v))$
Combined effect of $U$ and $T$ always satisfies linear transformation, then it is true.
This solution is very poor, so I want to find correct solution. How can I fix it or solve?
To be more vigorous you should say "for all scalars c,d,e,f in R and vectors u,v in R^n" and pick a different pair for doing U over R^m, though you may want to generalise to arbitrary vector fields for a more complete proof.