Vector space

There are two operations: addition of two vectors and multiplication of a vector by a scalar. The satisfy the following axioms:

1. X+Y=Y+X for all X and Y
2. X+(Y+Z)=(X+Y)+Z for all X, Y, and Z
3. X+0=X=0+X for some element 0 in V and all X
4. X+(-X)=0 for any X and some element -X in V
5. a(bX)=(ab)X for all X and all scalars a, b
6. 1X=X for all X
7. a(X+Y)=aX+aY for all X and Y
8. (a+b)X=aX+bX for all X, Y and all scalars a, b

See also Vectors.