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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)](https://tex.z-dn.net/?f=%5B%281%2C1%29%5Cbigoplus%282%2C2%29%5D%5Cbigoplus%283%2C3%29%3D%286%2C6%29%20)
![[(1,1)\bigoplus[(2,2)]\bigoplus(3,3)]=(6,6)](https://tex.z-dn.net/?f=%5B%281%2C1%29%5Cbigoplus%5B%282%2C2%29%5D%5Cbigoplus%283%2C3%29%5D%3D%286%2C6%29)
<span>
It's trivial to see this one holds.
2)</span>Commutativity<span> of addition
</span>
<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>
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.
5)Compatibility<span> of scalar multiplication with field multiplication
</span>This one holds.
![a[b(c,d)]=ab(c,d)](https://tex.z-dn.net/?f=a%5Bb%28c%2Cd%29%5D%3Dab%28c%2Cd%29)
6)<span>Identity element of scalar multiplication
</span>Identity element of scalar multiplication is simply 1.
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)](https://tex.z-dn.net/?f=a%5B%28b%2Cc%29%5Cbigoplus%28d%2Ce%29%5D%3Da%28b%2Cc%29%5Cbigoplus%20a%28d%2Ce%29)
Now let's look at the example:
![a[(1,1)\bigoplus(2,2)]=a(3,3)=(3,3a)](https://tex.z-dn.net/?f=a%5B%281%2C1%29%5Cbigoplus%282%2C2%29%5D%3Da%283%2C3%29%3D%283%2C3a%29)
This one hold too.
8)<span>Distributivity of scalar multiplication with respect to field addition
</span>Definition of this one is:
Let's take a look at the example:
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.
1) Associativity<span> of addition
</span>
It's trivial to see this one holds.
2)</span>Commutativity<span> of addition
</span>
<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>
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.
5)Compatibility<span> of scalar multiplication with field multiplication
</span>This one holds.
6)<span>Identity element of scalar multiplication
</span>Identity element of scalar multiplication is simply 1.
7)Distributivity of scalar multiplication with respect to vector addition. Let's look at the definition.
Now let's look at the example:
This one hold too.
8)<span>Distributivity of scalar multiplication with respect to field addition
</span>Definition of this one is:
Let's take a look at the example:
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.