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QUESTION: I don't understand what they are looking for about subset. Ay help would be greatly appreciated.

Explain why the following subsets are NOT subspace by giving a counter-example to one of the three properties.

The subsets can be seen in the picture.

ANSWER: A property of subspaces is that, if U1 and U2 are vectors in the subspace, then U1 + U2 = U3 is a vector in the subspace too. The first vector represents vectors for which V = (x, y, x-y=6); for any 2 vectors with this relationship between components

V1 + V2 = (x1+x2, x1-6+x2-6) = (x1+x2,x1+x2-12) ≠ (x1+x2, x1+x2-6) so these vector do not represent a subspace.

By writing out the sum of W1 and W2 in the same way you can show that W3 ≠ W1 + W2.

---------- FOLLOW-UP ----------

QUESTION: part a)

V={[x],[y] | x-y=6}

u and w is element of v such that u=(7,1) and w=(8,2)

u+w=(15,3)

u+w is not in V ,that to say V is not closed.

So it is not subspace.

for part b)

W={[[a],[b],[c]]| ab=c}

u and v are in W ,such that u=(1,2,2) ,v=(2,3,6)

u+v=(3,5,8)

W is not closed under addition .closure property does not hold.

So it is not subspace.

Are these the right ways of doing the problems? I fi am missing any key information, help would be appreciated.

Your derivations are fine as long as you really understand what not being "closed" means. Any set that is defined such that its elements don't obey the addition property does not constitute a subspace. BTW, I think part a) defines an affine transformation vis a linear one.

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