Let U be the set of all circles in R^2 having center the origin. Interpret the origin as being in this set; a circle centered at the origin with radius zero. Let C_1 and C_2 be elements of U. Let C_1 + C_2 be the circle centered at the origin, whose radius is the sum of the radii of C_1 and C_2. Let kC_1 be the circle centered at the origin, whose radius is |k| times that of C_1. The set U is not a vector space. Determine which vector space axioms (listed below) hold and which do not. For each axiom that you claim does not hold, provide justification. 1. Closed under addition; that is, the sum u + v exists and is an element of U.
2. Closed under scalar multiplication; that is, cu is an element of U.
3. u + v = v + u
4. u + (v + w) = (u + v) + w
5. There exists an element of U, called a zero vector, denoted 0, such that u + 0 = u.
6. For every element u of U, there exists an element called a negative of u, denoted -u, such that u + (-u) = 0.
7. c(u + v) = cu + cv
8. (c + d)u = cu + du
9. c(du) = (cd)u
10. 1u = u