Question

The given set together with the given operations is not a vector space. list the properties of the definition that fail to hold. (select all that apply.) the set of all ordered pairs of real numbers with the operations (x, y) ⊕ (x', y') = (x + x', y + y') and r (x, y) = (x, ry).

Answer 1

Let us look at all the axioms of a vector space and see which axioms are broken.
1) Associativity of addition

`[(1,1)\bigoplus(2,2)]\bigoplus(3,3)=(6,6)`

`[(1,1)\bigoplus[(2,2)]\bigoplus(3,3)]=(6,6)`
It's trivial to see this one holds.
2)Commutativity of addition

`(1,1)\bigoplus(2,2)=(2,2)\bigoplus(1,1)=(3,3)`
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).

`(a,b)\bigoplus(0,0)=(a,b)`
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.

`(a,b)\bigoplus(-a,-b)=(a-a,b-b)=0`
5)Compatibility of scalar multiplication with field multiplication
This one holds.

`a[b(c,d)]=ab(c,d)`
6)Identity element of scalar multiplication
Identity element of scalar multiplication is simply 1.

`1\cdot (a,b)=(a,1\cdot b)=(a,b)`
7)Distributivity of scalar multiplication with respect to vector addition. Let's look at the definition.

`a[(b,c)\bigoplus(d,e)]=a(b,c)\bigoplus a(d,e)`
Now let's look at the example:

`a[(1,1)\bigoplus(2,2)]=a(3,3)=(3,3a)`

`a(1,1)\bigoplus a(2,2)=(1,1a)\bigoplus(2,2a)=(3,3a)`
This one hold too.
8)Distributivity of scalar multiplication with respect to field addition
Definition of this one is:

`(a+b)(c,d)=a(c,d)+b(c,d)`
Let's take a look at the example:

`(3+2)(1,2)=5(1,2)=(1,10)`

`(3+2)(1,2)=3(1,2)+2(1,2)=(1,6)+(1,4)=(2,10)`
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.