Determine whether the given se, together with the soec?ied operat ons of addit i

Question

Determine whether the given se, together with the soec?ied operat ons of addit ion and scalar multiplication, is a vector space. I' it is nat, select all af the axiams that fail to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.) The set of al vectors in R with xy20 (i.e, the union of the fist and third quadrants), with the usual vector addition and scalar multiolicatian All of the axioms hold, so the given set is a vector space. -4. There exists an element? in V, called a zero vector, such that u + 0 = u. 5. For each u in V,there is an el ement -u in V such that u+[-u-0 6, curs in V.

Explanation / Answer

The sixth axiom doesn't follows.

Example c=-1

X>0

Then cx=(-1)x<0 which contradicts that xy>=0 for all x and y.

Thus, scalar multiplication fails. Hence it is not a vector space.

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