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AveGali [126]
3 years ago
5

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).
Mathematics
1 answer:
Triss [41]3 years ago
8 0
Let us look at all the axioms of a vector space and see which axioms are broken. 
1) Associativity<span> of addition
</span>[(1,1)\bigoplus(2,2)]\bigoplus(3,3)=(6,6)
[(1,1)\bigoplus[(2,2)]\bigoplus(3,3)]=(6,6)<span>
It's trivial to see this one holds.
2)</span>Commutativity<span> of addition
</span>(1,1)\bigoplus(2,2)=(2,2)\bigoplus(1,1)=(3,3)
<span>This one holds.
3)</span>Identity element<span> of addition
</span><span>This means that we have zero vektor. It is pretty obvious we do, it is (0,0). 
</span>(a,b)\bigoplus(0,0)=(a,b)
4)Inverse elements<span> of addition
</span>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<span> of scalar multiplication with field multiplication
</span>This one holds.
a[b(c,d)]=ab(c,d)
6)<span>Identity element of scalar multiplication
</span>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)<span>Distributivity of scalar multiplication with respect to field addition
</span>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.

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