Today I was studying linear transformations of vector spaces and ran across the following theorem:
Theorem: Let $T:V \rightarrow W$ be a linear transformation from a vector space $V$ to another vector space $W$. Let $0_v$ and $0_w$ be the zero vectors of $V$ and $W$ respectively. Then,
$$T(0_v)=0_w$$
Proof: Consider $v \in V$. Then $$T(v)=T(v+0_v)=T(v)+T(0_v)$$
From here we get-
$$T(0_v)=T(v)-T(v)=0_w$$
The last step seems rather strange to me. Can we use the rules of normal addition and subtraction over vector spaces like this? Isn't the definition of the zero vector dependent on what the rules for addition and multiplication in it are? And even if they are different from ordinary rules, can the above theorem be as general as the author claims it to be?
Best Answer
By definition of vector space: $W$ forms a commutative group with respect to "addition". Here $0_w$ is the identity of that group and $-T(v)$ is denoted as the inverse of $T(v)$ with respect to that specially defined operation "addition". Now you just put $-T(v)$ to both sides of the first equation and use group operation.
Hope it helps.