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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?

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  • $\begingroup$ @angryavian linear transformation $\endgroup$ – JOHN Aug 28 at 0:13
  • $\begingroup$ linear transformations with addition and scalar multiplication comprise a vector space $\endgroup$ – J. W. Tanner Aug 28 at 0:18
  • $\begingroup$ The book also does not state that a linear transformation is a vector space. Rather, it is the set of all linear transformations from $V$ to $W$ which is a vector space with the given operations. (Here $V$ and $W$ are vector spaces over a field $F$.) You do indeed need to check the vector space axioms one by one to confirm that they are satisfied. $\endgroup$ – littleO Aug 28 at 0:18
  • $\begingroup$ The quoted section doesn't appear to be intended as a proof of anything. It appears to be a comment on something the author thinks is already proved, saying that the new fact should not be a complete surprise to you if you have been paying attention. As to how rigorous the preceding proof was, we would need to see it (and possibly also to see a number of earlier results that might have given you ways to verify a vector space other than literally listing and checking the axioms one at a time) in order to decide that. $\endgroup$ – David K Aug 28 at 0:52
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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.

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