Mathematics — Linear Algebra · vector-space-axioms
Problem
Define a possibly new vector space by beginning with the set and vector addition from C2 (Example VSCV) but change the definition of scalar multiplication to
αx=0=[00]α∈C,x∈C2
Prove that the first nine properties required for a vector space hold, but Property O does not hold.
This example shows us that we cannot expect to be able to derive Property O as a consequence of assuming the first nine properties. In other words, we cannot slim down our list of properties by jettisoning the last one, and still have the same collection of objects qualify as vector spaces.