Let us look at all the axioms of a vector space and see which axioms are broken.
1) Associativity of addition
[tex][(1,1)\bigoplus(2,2)]\bigoplus(3,3)=(6,6) [/tex]
[tex][(1,1)\bigoplus[(2,2)]\bigoplus(3,3)]=(6,6)[/tex]
It's trivial to see this one holds.
2)Commutativity of addition
[tex](1,1)\bigoplus(2,2)=(2,2)\bigoplus(1,1)=(3,3)[/tex]
This one holds.
3)Identity element of addition
This means that we have zero vektor. It is pretty obvious we do, it is (0,0).
[tex](a,b)\bigoplus(0,0)=(a,b)[/tex]
4)Inverse elements of addition
This means that for each element in V there exists element -v such that v+(-v)=0.
We do have inverse elements.
[tex](a,b)\bigoplus(-a,-b)=(a-a,b-b)=0[/tex]
5)Compatibility of scalar multiplication with field multiplication
This one holds.
[tex]a[b(c,d)]=ab(c,d)[/tex]
6)Identity element of scalar multiplication
Identity element of scalar multiplication is simply 1.
[tex]1\cdot (a,b)=(a,1\cdot b)=(a,b)[/tex]
7)Distributivity of scalar multiplication with respect to vector addition. Let's look at the definition.
[tex]a[(b,c)\bigoplus(d,e)]=a(b,c)\bigoplus a(d,e)[/tex]
Now let's look at the example:
[tex]a[(1,1)\bigoplus(2,2)]=a(3,3)=(3,3a)[/tex]
[tex]a(1,1)\bigoplus a(2,2)=(1,1a)\bigoplus(2,2a)=(3,3a)[/tex]
This one hold too.
8)Distributivity of scalar multiplication with respect to field addition
Definition of this one is:
[tex](a+b)(c,d)=a(c,d)+b(c,d)[/tex]
Let's take a look at the example:
[tex](3+2)(1,2)=5(1,2)=(1,10)[/tex]
[tex](3+2)(1,2)=3(1,2)+2(1,2)=(1,6)+(1,4)=(2,10)[/tex]
So this one doesn't hold.
The final answer would be distributivity of scalar multiplication with respect to field addition.
Please note vectors, in this case, are (a,b) and that I did not use the dot to indicate scalar multiplication.
