Consider R2 with the following rules of multiplications and additions: For each x=(x1,x2), y=(y1,y2):
x+y=(x2+y2,x1+y1) and for any scalar alpha, alpha*x=(alpha*x1, alpha*x2)
Here is the definition of a VECTOR SPACE:
There is an addition '+' in V such that V,+ is a commutative group.
Any element v in V and any r in R determine a scalar product rv in V.
This scalar product has the following properties for any r,s in R and any v,w in V. ...
Mulitplication and addition rules are investigated for vectors and scalars.