Assume:

If <a,b> and <c,d> are vectors in the vector spaceR², wherea,b,c, anddare real numbers andris a scalar, then the following operations are defined:

vector addition is defined as <a,b> + <c,d>=<a+c,b+d>

scalar multiplication is defined asr<a,b>=<ra,rb>

Note: There are various acceptable notations for a vector:<a,b> = [a,b] = (a,b) =[_b^a]

Problem:

A vector space,V, with vectorsX, Y, andZsatisfies the following 10 laws, whererandsare real number scalars:

law 1: closure under addition

IfXandYare any two vectors inV, thenX+YV.

law 2: associativity of vector addition

(X+Y) +Z=X+ (Y+Z)

law 3: commutative law

X+Y=Y+X

law 4: additive identity law

There is a vector inV, denoted by0such thatX+0=Xwhere0is called the zero vector.

law 5: additive inverse law

For everyXVthere is a vector -Xsuch thatX+ (-X) =0, where -Xis called the additive inverse ofX.

law 6: closure under scalar multiplication

IfXis any vector inVandris any real scalar, thenrXV.

law 7: associativity of scalar multiplication

(rs)X=r(sX)

law 8: distributivity of vector sums

r(X+Y) =rX+rY

law 9: distributivity of scalar sums

(r+s)X=rX+sX

law 10: scalar multiplication identity

1X=X

Please help me understand, almost step by step, I am lost. Help!