I am reading Linear Algebra Done Wrong.
In Section 4 Linear transformation as a vector space, after the Author defines the scalar multiplication and addition of transformations and proved that they are indeed linear, it said
This (operations satisfy axioms of vector space) should come as no surprise for the reader, since axioms of a vector space essentially mean that operation on vector follow standard rules of algebra. And the operations on linear transformations are defined as to satisfy these rules.
Normally, I will check the axioms one by one. It seems that by just looking at the defined operations of scalar multiplication, and addition, we can know immediate that it is a vector space.
How to do you think about it or convince yourself?