Proof of Theorem VSLT

Theorem SLTLT and Theorem MLTLT provide two of the ten properties in Definition VS. However, we still need to verify the remaining eight properties. By and large, the proofs are straightforward and rely on concocting the obvious object, or by reducing the question to the same vector space property in the vector space $V$.

The zero vector is of some interest, though. What linear transformation would we add to any other linear transformation, so as to keep the second one unchanged? The answer is $\ltdefn{Z}{U}{V}$ defined by $\lt{Z}{\vect{u}}=\zerovector_V$ for every $\vect{u}\in U$. Notice how we do not need to know any of the specifics about $U$ and $V$ to make this definition of $Z$.