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?
Best Answer
On the level of rigorous proof: You still have to check the axioms one by one (note that the author checks one of the axioms immediately afterward).
On an intuitive level: the operation of adding linear transformations just means to add their outputs, and the operation of scaling a linear transformation just means to scale its output. Since these additions and scalings as operations on the outputs satisfy all the vector space outcomes, the operations on the transformations should as well. This should become more intuitively clear if you work through the proofs of some of the axioms.