## UnkleRhaukus 3 years ago What is the difference between a vector space and a set.

1. satellite73

a raft of axioms that have to be satisfied, as well as a couple of operations

2. myko

a set in general is just a collection of "objects" with or without any condition imposed on the members of the set. Vector space is also kind of set, but with special condition to be satisfied by the elements that belong to it.

3. myko

conditions*

4. UnkleRhaukus

which conditions, axioms, operations, are usually not common to vector spaces and sets

5. myko
6. UnkleRhaukus

is the set of complex numbers a vector space?

7. myko

yes

8. UnkleRhaukus

what about the set on n dimensional matrices ?

9. UnkleRhaukus

n dimensional complex matrices ?

10. myko

also

11. UnkleRhaukus

are you sure

12. UnkleRhaukus

*^ n by n complex matrices ?

13. cumin

can you add 2 by n complex matrices and get an n by n complex matrix? can you multiply an n by n complex matrix by a scalar (a real number would be a simple scalar to use here maybe) and get an n by n complex matrix as a result? If the answer is yes to both questions, I think that means that n by n complex matrices make up a vector space. If I am wrong I would like to be corrected.